{"id":15494,"date":"2019-09-05T12:07:42","date_gmt":"2019-09-05T16:07:42","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/ellipses-2\/"},"modified":"2019-09-05T12:07:42","modified_gmt":"2019-09-05T16:07:42","slug":"ellipses-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/ellipses-2\/","title":{"raw":"Ellipses","rendered":"Ellipses"},"content":{"raw":"[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Graph an ellipse with center at the origin<\/li><li>Find the equation of an ellipse with center at the origin<\/li><li>Graph an ellipse with center not at the origin<\/li><li>Solve application with ellipses<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1163872103152\" class=\"be-prepared\"><p id=\"fs-id1163872387647\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1163872108629\" type=\"1\"><li>Graph \\(y={\\left(x-1\\right)}^{2}-2\\) using transformations.<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/9054900c-f191-40ed-8ceb-e4f051bbbf2b#fs-id1169148912189\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Complete the square: \\({x}^{2}-8x=8.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/3e7f365d-4885-4b7a-bb2a-4358cbc00d2e#fs-id1167829894368\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Write in standard form. \\(y=2{x}^{2}-12x+14\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/9054900c-f191-40ed-8ceb-e4f051bbbf2b#fs-id1169149374763\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872520186\"><h3 data-type=\"title\">Graph an Ellipse with Center at the Origin<\/h3><p id=\"fs-id1163872416455\">The next conic section we will look at is an <span data-type=\"term\">ellipse<\/span>. We define an ellipse as all points in a plane where the sum of the distances from two fixed points is constant. Each of the given points is called a <strong data-effect=\"bold\">focus<\/strong> of the ellipse.<\/p><div data-type=\"note\" id=\"fs-id1163872730452\"><div data-type=\"title\">Ellipse<\/div><p id=\"fs-id1163872721606\">An <strong data-effect=\"bold\">ellipse<\/strong> is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the ellipse.<\/p><span data-type=\"media\" id=\"fs-id1163872515324\" data-alt=\"This figure shows a double cone intersected by a plane to form an ellipse.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a double cone intersected by a plane to form an ellipse.\" \/><\/span><\/div><p id=\"fs-id1163872009594\">We can draw an ellipse by taking some fixed length of flexible string and attaching the ends to two thumbtacks. We use a pen to pull the string taut and rotate it around the two thumbtacks. The figure that results is an ellipse.<\/p><span data-type=\"media\" id=\"fs-id1163871857451\" data-alt=\"This figure shows a pen attached to two strings, the other ends of which are attached to two thumbtacks. The strings are pulled taut and the pen is rotated to draw an ellipse. The thumbtacks are labeled F subscript 1 and F subscript 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a pen attached to two strings, the other ends of which are attached to two thumbtacks. The strings are pulled taut and the pen is rotated to draw an ellipse. The thumbtacks are labeled F subscript 1 and F subscript 2.\" \/><\/span><p id=\"fs-id1163868232249\">A line drawn through the foci intersect the ellipse in two points. Each point is called a <strong data-effect=\"bold\">vertex<\/strong> of the ellipse. The segment connecting the vertices is called the <strong data-effect=\"bold\">major axis<\/strong>. The midpoint of the segment is called the <strong data-effect=\"bold\">center<\/strong> of the ellipse. A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the <strong data-effect=\"bold\">minor axis<\/strong>.<\/p><span data-type=\"media\" id=\"fs-id1163866197380\" data-alt=\"This figure shows two ellipses. In each, two points within the ellipse are labeled foci. A line drawn through the foci intersects the ellipse in two points. Each point is labeled a vertex. In The figure on the left, the segment connecting the vertices is called the major axis. A segment perpendicular to the major axis that passes through its midpoint and intersects the ellipse in two points is labeled minor axis. The major axis is longer than the minor axis. In The figure on the right, the segment through the foci, connecting the vertices is shorter and is labeled minor axis. Its midpoint is labeled center.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two ellipses. In each, two points within the ellipse are labeled foci. A line drawn through the foci intersects the ellipse in two points. Each point is labeled a vertex. In The figure on the left, the segment connecting the vertices is called the major axis. A segment perpendicular to the major axis that passes through its midpoint and intersects the ellipse in two points is labeled minor axis. The major axis is longer than the minor axis. In The figure on the right, the segment through the foci, connecting the vertices is shorter and is labeled minor axis. Its midpoint is labeled center.\" \/><\/span><p id=\"fs-id1163871867164\">We mentioned earlier that our goal is to connect the geometry of a conic with algebra. Placing the ellipse on a rectangular coordinate system gives us that opportunity. In the figure, we placed the ellipse so the foci \\(\\left(\\left(\\text{\u2212}c,0\\right),\\left(c,0\\right)\\right)\\) are on the <em data-effect=\"italics\">x<\/em>-axis and the center is the origin.<\/p><span data-type=\"media\" id=\"fs-id1163872705598\" data-alt=\"The figure on the left shows an ellipse with its center at the origin of the coordinate axes and its foci at points minus (c, 0) and (c, 0). A segment connects (negative c, 0) to a point (x, y) on the ellipse. The segment is labeled d subscript 1. Another segment, labeled d subscript 2 connects (c, 0) to (x, y). The figure on the right shows an ellipse with center at the origin, foci (negative c, 0) and (c, 0) and vertices (negative a, 0) and (a, 0). The point where the ellipse intersects the y axis is labeled (0, b). The segments connecting (0, 0) to (c, 0), (c, 0) to (0, b) and (0, b) to (0, 0) form a tight angled triangle with sides c, a and b respectively. The equation is a squared equals b squared plus c squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure on the left shows an ellipse with its center at the origin of the coordinate axes and its foci at points minus (c, 0) and (c, 0). A segment connects (negative c, 0) to a point (x, y) on the ellipse. The segment is labeled d subscript 1. Another segment, labeled d subscript 2 connects (c, 0) to (x, y). The figure on the right shows an ellipse with center at the origin, foci (negative c, 0) and (c, 0) and vertices (negative a, 0) and (a, 0). The point where the ellipse intersects the y axis is labeled (0, b). The segments connecting (0, 0) to (c, 0), (c, 0) to (0, b) and (0, b) to (0, 0) form a tight angled triangle with sides c, a and b respectively. The equation is a squared equals b squared plus c squared.\" \/><\/span><p id=\"fs-id1163871890058\">The definition states the sum of the distance from the foci to a point \\(\\left(x,y\\right)\\) is constant. So \\({d}_{1}+{d}_{2}\\) is a constant that we will call \\(2a\\) so, \\({d}_{1}+{d}_{2}=2a.\\) We will use the distance formula to lead us to an algebraic formula for an ellipse.<\/p><div data-type=\"equation\" id=\"fs-id1163872419619\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\begin{array}{}\\\\ \\\\ \\\\ \\\\ \\\\ \\text{Use the distance formula to find}\\phantom{\\rule{0.2em}{0ex}}{d}_{1},{d}_{2}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\begin{array}{ccccc}\\hfill {d}_{1}\\hfill &amp; +\\hfill &amp; \\hfill {d}_{2}\\hfill &amp; =\\hfill &amp; 2a\\hfill \\\\ \\\\ \\\\ \\sqrt{{\\left(x-\\left(\\text{\u2212}c\\right)\\right)}^{2}+{\\left(y-0\\right)}^{2}}\\hfill &amp; +\\hfill &amp; \\sqrt{{\\left(x-c\\right)}^{2}+{\\left(y-0\\right)}^{2}}\\hfill &amp; =\\hfill &amp; 2a\\hfill \\end{array}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{After eliminating radicals and simplifying,}\\hfill \\\\ \\text{we get:}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{a}^{2}-{c}^{2}}\\phantom{\\rule{0.5em}{0ex}}=\\phantom{\\rule{0.5em}{0ex}}1\\phantom{\\rule{0.6em}{0ex}}\\\\ \\begin{array}{c}\\text{To simplify the equation of the ellipse, we}\\hfill \\\\ \\text{let}\\phantom{\\rule{0.2em}{0ex}}{a}^{2}-{c}^{2}={b}^{2}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\begin{array}{c}\\text{So, the equation of an ellipse centered at the}\\hfill \\\\ \\text{origin in standard form is:}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}\\phantom{\\rule{0.5em}{0ex}}=\\phantom{\\rule{0.5em}{0ex}}1\\phantom{\\rule{0.6em}{0ex}}\\end{array}\\)<\/div><p id=\"fs-id1163871567318\">To graph the ellipse, it will be helpful to know the intercepts. We will find the <em data-effect=\"italics\">x<\/em>-intercepts and <em data-effect=\"italics\">y<\/em>-intercepts using the formula.<\/p><div data-type=\"equation\" id=\"fs-id1163872646544\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{0.5em}{0ex}}{\\text{y}}\\mathbf{\\text{-intercepts}}\\hfill \\\\ \\begin{array}{c}\\text{Let}\\phantom{\\rule{0.2em}{0ex}}x=0.\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{ccc}\\hfill \\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill \\frac{{0}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill \\frac{{y}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill {y}^{2}&amp; =\\hfill &amp; {b}^{2}\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; \u00b1b\\hfill \\end{array}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{0.5em}{0ex}}{\\text{x}}\\mathbf{\\text{-intercepts}}\\hfill \\\\ \\begin{array}{c}\\text{Let}\\phantom{\\rule{0.2em}{0ex}}y=0.\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{ccc}\\hfill \\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill \\frac{{x}^{2}}{{a}^{2}}+\\frac{{0}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill \\frac{{x}^{2}}{{a}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill {x}^{2}&amp; =\\hfill &amp; {a}^{2}\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; \u00b1a\\hfill \\end{array}\\hfill \\end{array}\\hfill \\\\ \\text{The}\\phantom{\\rule{0.2em}{0ex}}y\\text{-intercepts are}\\phantom{\\rule{0.2em}{0ex}}\\left(0,b\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\left(0,\\text{\u2212}b\\right).\\hfill &amp; &amp; &amp; \\text{The}\\phantom{\\rule{0.2em}{0ex}}x\\text{-intercepts are}\\phantom{\\rule{0.2em}{0ex}}\\left(a,0\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\left(\\text{\u2212}a,0\\right).\\hfill \\end{array}\\)<\/div><div data-type=\"note\" id=\"fs-id1163871935632\"><div data-type=\"title\">Standard Form of the Equation an Ellipse with Center \\(\\left(0,\\text{\u200b}\\text{\u200b}0\\right)\\)<\/div><p id=\"fs-id1163872096452\">The standard form of the equation of an ellipse with center \\(\\left(0,\\text{\u200b}\\text{\u200b}0\\right),\\) is<\/p><div data-type=\"equation\" id=\"fs-id1163872630602\" class=\"unnumbered\" data-label=\"\">\\(\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\)<\/div><p id=\"fs-id1163868452315\">The <em data-effect=\"italics\">x<\/em>-intercepts are \\(\\left(a,0\\right)\\) and \\(\\left(\\text{\u2212}a,0\\right).\\)<\/p><p id=\"fs-id1163872541971\">The <em data-effect=\"italics\">y<\/em>-intercepts are \\(\\left(0,b\\right)\\) and \\(\\left(0,\\text{\u2212}b\\right).\\)<\/p><span data-type=\"media\" id=\"fs-id1163872938165\" data-alt=\"Two figures show ellipses with their centers on the origin of the coordinate axes. They intersect the x axis at points (negative a, 0) and (a, 0) and the y axis at points (0, b) and (0, negative b). In the figure on the left the major axis of the ellipse is along the x axis and in the figure on the right, it is along the y axis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_005_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two figures show ellipses with their centers on the origin of the coordinate axes. They intersect the x axis at points (negative a, 0) and (a, 0) and the y axis at points (0, b) and (0, negative b). In the figure on the left the major axis of the ellipse is along the x axis and in the figure on the right, it is along the y axis.\" \/><\/span><\/div><p id=\"fs-id1163871911233\">Notice that when the major axis is horizontal, the value of <em data-effect=\"italics\">a<\/em> will be greater than the value of <em data-effect=\"italics\">b<\/em> and when the major axis is vertical, the value of <em data-effect=\"italics\">b<\/em> will be greater than the value of <em data-effect=\"italics\">a<\/em>. We will use this information to graph an <span data-type=\"term\" class=\"no-emphasis\">ellipse<\/span> that is centered at the origin.<\/p><table id=\"fs-id1163872513141\" summary=\".\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Ellipse with Center \\(\\left(0,0\\right)\\)<\/th><\/tr><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\)<\/th><th data-valign=\"middle\" data-align=\"center\">\\(a&gt;b\\)<\/th><th data-valign=\"middle\" data-align=\"center\">\\(b&gt;a\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Major axis<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">on the <em data-effect=\"italics\">x<\/em>- axis.<\/td><td data-valign=\"middle\" data-align=\"center\">on the <em data-effect=\"italics\">y<\/em>-axis.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td><td colspan=\"2\" data-valign=\"middle\" data-align=\"center\">\\(\\left(\\text{\u2212}a,0\\right),\\)\\(\\left(a,0\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td><td colspan=\"2\" data-valign=\"middle\" data-align=\"center\">\\(\\left(0,\\text{\u2212}b\\right),\\)\\(\\left(0,b\\right)\\)<\/td><\/tr><\/tbody><\/table><div data-type=\"example\" id=\"fs-id1163872731080\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Graph an Ellipse with Center (0, 0)<\/div><div data-type=\"exercise\" id=\"fs-id1163868248185\"><div data-type=\"problem\" id=\"fs-id1163872022977\"><p id=\"fs-id1163872468441\">Graph: \\(\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872538228\"><span data-type=\"media\" id=\"fs-id1163872427329\" data-alt=\"Step 1. Write the equation in standard form. It is in standard form x squared upon 6 plus y squared upon 9 equals 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1. Write the equation in standard form. It is in standard form x squared upon 6 plus y squared upon 9 equals 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163872399906\" data-alt=\"Step 2. Determine whether the major axis is horizontal or vertical. Since 9 is greater than 4 and 9 is in the y squared term, the major axis is vertical.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. Determine whether the major axis is horizontal or vertical. Since 9 is greater than 4 and 9 is in the y squared term, the major axis is vertical.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163872642709\" data-alt=\"Step 3. Find the endpoints of the major axis. The endpoints will be the y-intercepts. Since b squared is 9, b is plus or minus 3. The endpoints of the major axis are (0, 3) and (0, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. Find the endpoints of the major axis. The endpoints will be the y-intercepts. Since b squared is 9, b is plus or minus 3. The endpoints of the major axis are (0, 3) and (0, negative 3).\" \/><\/span><span data-type=\"media\" id=\"fs-id1163871870410\" data-alt=\"Step 4. Find the endpoints of the minor axis. The endpoints will be the x-intercepts. Since a squared is 4, a is plus or minus 2. The endpoints of the minor axis are (2, 0) and (negative 2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Find the endpoints of the minor axis. The endpoints will be the x-intercepts. Since a squared is 4, a is plus or minus 2. The endpoints of the minor axis are (2, 0) and (negative 2, 0).\" \/><\/span><span data-type=\"media\" id=\"fs-id1163872842289\" data-alt=\"Step 5. Sketch the ellipse using the x and y intercepts. The graph shows an ellipse with center at (0, 0) and foci at (0, 3), (0, negative 3), (negative 2, 0), and (2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5. Sketch the ellipse using the x and y intercepts. The graph shows an ellipse with center at (0, 0) and foci at (0, 3), (0, negative 3), (negative 2, 0), and (2, 0).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872473001\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872449228\"><div data-type=\"problem\" id=\"fs-id1163872654950\"><p id=\"fs-id1163872435556\">Graph: \\(\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{16}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872667846\"><span data-type=\"media\" id=\"fs-id1163872543638\" data-alt=\"This graph shows an ellipse with x intercepts (negative 2, 0) and (2, 0) and y intercepts (0, 4) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 2, 0) and (2, 0) and y intercepts (0, 4) and (0, negative 4).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872560939\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872515144\"><div data-type=\"problem\" id=\"fs-id1163871870306\"><p id=\"fs-id1163872727725\">Graph: \\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{16}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872470405\"><span data-type=\"media\" id=\"fs-id1163871979084\" data-alt=\"This graph shows an ellipse with x intercepts (negative 3, 0) and (3, 0) and y intercepts (0, 4) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 3, 0) and (3, 0) and y intercepts (0, 4) and (0, negative 4).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163872513225\">We summarize the steps for reference.<\/p><div data-type=\"note\" id=\"fs-id1163872664715\" class=\"howto\"><div data-type=\"title\">How to Graph an Ellipse with Center \\(\\left(0,0\\right).\\)<\/div><ol id=\"fs-id1163872554151\" type=\"1\" class=\"stepwise\"><li>Write the equation in standard form.<\/li><li>Determine whether the major axis is horizontal or vertical.<\/li><li>Find the endpoints of the major axis.<\/li><li>Find the endpoints of the minor axis<\/li><li>Sketch the ellipse.<\/li><\/ol><\/div><p id=\"fs-id1163871890228\">Sometimes our equation will first need to be put in standard form.<\/p><div data-type=\"example\" id=\"fs-id1163872427815\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872611858\"><div data-type=\"problem\" id=\"fs-id1163872539321\"><p id=\"fs-id1163872391526\">Graph \\({x}^{2}+4{y}^{2}=16.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163868452816\"><table id=\"fs-id1163872012967\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x squared plus 4 y squared equals 16. We recognize this as the equation of an ellipse since both the x and y terms are squared and have different coefficients. To get the equation in standard form, divide both sides by 16 so that the right side of the equation is equal to 1. Simplify to get x squared upon 16 plus y squared upon 4 equals 1. The equation is in standard form. The ellipse is centered at the origin, (0, 0). Since 16 is greater than 4 and 16 is in the x squared term, the major axis is horizontal. a squared is 16, so a is plus or minus 4. The vertices are (4, 0) and (negative 4, 0). b squared is 4, so b is plus or minus 2. The endpoints of the minor axis are (0, 2) and (0, negative 2). Sketch the ellipse.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">We recognize this as the equation of an<span data-type=\"newline\"><br \/><\/span>ellipse since both the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> terms are<span data-type=\"newline\"><br \/><\/span>squared and have different coefficients.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{0.9em}{0ex}}{x}^{2}+4{y}^{2}=16\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To get the equation in standard form, divide<span data-type=\"newline\"><br \/><\/span>both sides by 16 so that the equation is equal<span data-type=\"newline\"><br \/><\/span>to 1.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{0.61em}{0ex}}\\frac{{x}^{2}}{16}+\\frac{4{y}^{2}}{16}=\\frac{16}{16}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{0.67em}{0ex}}\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{4}=1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The equation is in standard form.<span data-type=\"newline\"><br \/><\/span>The ellipse is centered at the origin.<\/td><td data-valign=\"top\" data-align=\"center\">The center is \\(\\left(0,0\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(16&gt;4\\) and 16 is in the \\({x}^{2}\\) term,<span data-type=\"newline\"><br \/><\/span>the major axis is horizontal.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2003\u2003\\({a}^{2}=16,a=\u00b14\\)<span data-type=\"newline\"><br \/><\/span>\u2003\u2003\\({b}^{2}=4,\\phantom{\\rule{0.55em}{0ex}}b=\u00b12\\)<\/td><td data-valign=\"top\" data-align=\"left\">The vertices are \\(\\left(4,0\\right),\\left(-4,0\\right).\\)<span data-type=\"newline\"><br \/><\/span>The endpoints of the minor axis are<span data-type=\"newline\"><br \/><\/span>\\(\\left(0,2\\right),\\left(0,-2\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the parabola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872562489\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872840418\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872388297\"><div data-type=\"problem\" id=\"fs-id1163872557543\"><p id=\"fs-id1163872425484\">Graph \\(9{x}^{2}+16{y}^{2}=144.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872446494\"><span data-type=\"media\" id=\"fs-id1163872542325\" data-alt=\"This graph shows an ellipse with x intercepts (negative 4, 0) and (4, 0) and y intercepts (0, 3) and (0, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 4, 0) and (4, 0) and y intercepts (0, 3) and (0, negative 3).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872691837\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872437330\"><div data-type=\"problem\" id=\"fs-id1163872016439\"><p id=\"fs-id1163872512504\">Graph \\(16{x}^{2}+25{y}^{2}=400.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872694456\"><span data-type=\"media\" id=\"fs-id1163871860728\" data-alt=\"This graph shows an ellipse with x intercepts (negative 5, 0) and (5, 0) and y intercepts (0, 4) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 5, 0) and (5, 0) and y intercepts (0, 4) and (0, negative 4).\" \/><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872654447\"><h3 data-type=\"title\">Find the Equation of an Ellipse with Center at the Origin<\/h3><p id=\"fs-id1163871865681\">If we are given the graph of an <span data-type=\"term\" class=\"no-emphasis\">ellipse<\/span>, we can find the equation of the ellipse.<\/p><div data-type=\"example\" id=\"fs-id1163868321690\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872829077\"><div data-type=\"problem\" id=\"fs-id1163872728911\"><p id=\"fs-id1163872714191\">Find the equation of the ellipse shown.<\/p><span data-type=\"media\" id=\"fs-id1163872544767\" data-alt=\"This graph shows an ellipse with x intercepts (negative 4, 0) and (4, 0) and y intercepts (0, 3) and (0, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_008_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 4, 0) and (4, 0) and y intercepts (0, 3) and (0, negative 3).\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872446354\"><p id=\"fs-id1163872420512\">\\(\\begin{array}{cccccc}\\begin{array}{c}\\text{We recognize this as an ellipse that is}\\hfill \\\\ \\text{centered at the origin.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\\\ \\begin{array}{c}\\text{Since the major axis is horizontal and the}\\hfill \\\\ \\text{distance from the center to the vertex is 4, we}\\hfill \\\\ \\text{know}\\phantom{\\rule{0.2em}{0ex}}a=4\\phantom{\\rule{0.2em}{0ex}}\\text{and so}\\phantom{\\rule{0.2em}{0ex}}{a}^{2}=16.\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{{b}^{2}}=1\\\\ \\begin{array}{c}\\text{The minor axis is vertical and the distance}\\hfill \\\\ \\text{from the center to the ellipse is 3, we know}\\hfill \\\\ b=3\\phantom{\\rule{0.2em}{0ex}}\\text{and so}\\phantom{\\rule{0.2em}{0ex}}{b}^{2}=9.\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1\\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163871618969\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872750117\"><div data-type=\"problem\" id=\"fs-id1163872836610\"><p id=\"fs-id1163872511100\">Find the equation of the ellipse shown.<\/p><span data-type=\"media\" id=\"fs-id1163872565747\" data-alt=\"This graph shows an ellipse with x intercepts (negative 2, 0) and (2, 0) and y intercepts (0, 5) and (0, negative 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_009_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 2, 0) and (2, 0) and y intercepts (0, 5) and (0, negative 5).\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872556813\"><p id=\"fs-id1163872638056\">\\(\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{25}=1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163871930788\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872547351\"><div data-type=\"problem\" id=\"fs-id1163868453510\"><p id=\"fs-id1163872607559\">Find the equation of the ellipse shown.<\/p><span data-type=\"media\" id=\"fs-id1163872690626\" data-alt=\"This graph shows an ellipse with x intercepts (negative 3, 0) and (3, 0) and y intercepts (0, 2) and (0, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_010_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 3, 0) and (3, 0) and y intercepts (0, 2) and (0, negative 2).\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163871980514\"><p id=\"fs-id1163872689651\">\\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{4}=1\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872533788\"><h3 data-type=\"title\">Graph an Ellipse with Center Not at the Origin<\/h3><p id=\"fs-id1163872643895\">The ellipses we have looked at so far have all been centered at the origin. We will now look at ellipses whose center is \\(\\left(h,k\\right).\\)<\/p><p id=\"fs-id1163872545176\">The equation is \\(\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1\\) and when \\(a&gt;b,\\) the major axis is horizontal so the distance from the center to the vertex is <em data-effect=\"italics\">a<\/em>. When \\(b&gt;a,\\) the major axis is vertical so the distance from the center to the vertex is <em data-effect=\"italics\">b<\/em>.<\/p><div data-type=\"note\" id=\"fs-id1163872729923\"><div data-type=\"title\">Standard Form of the Equation an Ellipse with Center \\(\\left(h,k\\right)\\) <\/div><p id=\"fs-id1163872447153\">The standard form of the equation of an ellipse with center \\(\\left(h,k\\right),\\) is<\/p><div data-type=\"equation\" id=\"fs-id1163872564491\" class=\"unnumbered\" data-label=\"\">\\(\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1\\)<\/div><p id=\"fs-id1163871867293\">When \\(a&gt;b,\\) the major axis is horizontal so the distance from the center to the vertex is <em data-effect=\"italics\">a<\/em>.<\/p><p id=\"fs-id1163871881869\">When \\(b&gt;a,\\) the major axis is vertical so the distance from the center to the vertex is <em data-effect=\"italics\">b<\/em>.<\/p><\/div><div data-type=\"example\" id=\"fs-id1163872427883\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872700016\"><div data-type=\"problem\" id=\"fs-id1163871999656\"><p id=\"fs-id1163872463704\">Graph: \\(\\frac{{\\left(x-3\\right)}^{2}}{9}+\\frac{{\\left(y-1\\right)}^{2}}{4}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872017878\"><table id=\"fs-id1163872729668\" class=\"unnumbered unstyled\" summary=\"The equation is open parentheses x minus 3 close parentheses squared upon 9 plus open parentheses y minus 1 close parentheses squared upon 4 equals 1. This is in standard form. The ellipse is centered at (h, k), which is (3, 1). Since 9 is greater than 4 and is in the x squared term, the major axis is horizontal. a squared is 9, so a is plus or minus 3. The distance from the center to the vertices is 3. b squared is 4, so b is plus or minus 2. The distance from the center to the endpoints of the minor axis is 2. Sketch the ellipse with point 3, 3, point 3, negative 3, point 6, 1) and (point 0, 1).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The equation is in standard form,<span data-type=\"newline\"><br \/><\/span>\\(\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\frac{{\\left(x-3\\right)}^{2}}{9}+\\frac{{\\left(y-1\\right)}^{2}}{4}=1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The ellipse is centered at \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">The center is \\(\\left(3,1\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(9&gt;4\\) and 9 is in the \\({x}^{2}\\) term,<span data-type=\"newline\"><br \/><\/span>the major axis is horizontal.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2003\u2003\\({a}^{2}=9,\\phantom{\\rule{0.25em}{0ex}}a=\u00b13\\)<span data-type=\"newline\"><br \/><\/span>\u2003\u2003\\({b}^{2}=4,\\phantom{\\rule{0.25em}{0ex}}b=\u00b12\\)<\/td><td data-valign=\"top\" data-align=\"left\">The distance from the center to the vertices is 3.<span data-type=\"newline\"><br \/><\/span>The distance from the center to the endpoints of the<span data-type=\"newline\"><br \/><\/span>minor axis is 2.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the ellipse.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871925897\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872516051\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872456046\"><div data-type=\"problem\" id=\"fs-id1163872403700\"><p id=\"fs-id1163872541264\">Graph: \\(\\frac{{\\left(x+3\\right)}^{2}}{4}+\\frac{{\\left(y-5\\right)}^{2}}{16}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872643398\"><span data-type=\"media\" id=\"fs-id1163871993543\" data-alt=\"This graph shows an ellipse with center at (negative 3, 5), vertices at (negative 3, 9) and (negative 3, 1) and endpoints of minor axis at (negative 5, 5) and (negative 1, 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center at (negative 3, 5), vertices at (negative 3, 9) and (negative 3, 1) and endpoints of minor axis at (negative 5, 5) and (negative 1, 5).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872769827\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163871859210\"><div data-type=\"problem\" id=\"fs-id1163871617684\"><p id=\"fs-id1163872545739\">Graph: \\(\\frac{{\\left(x-1\\right)}^{2}}{25}+\\frac{{\\left(y+3\\right)}^{2}}{16}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872503302\"><span data-type=\"media\" id=\"fs-id1163872652811\" data-alt=\"This graph shows an ellipse with center at 1, negative 3, vertices at (negative 4, negative 3) and (6, negative 3) and endpoints of minor axis at 1, 1) and (negative 1, negative 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_307_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center at 1, negative 3, vertices at (negative 4, negative 3) and (6, negative 3) and endpoints of minor axis at 1, 1) and (negative 1, negative 7).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163872033617\">If we look at the equations of \\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{4}=1\\) and \\(\\frac{{\\left(x-3\\right)}^{2}}{9}+\\frac{{\\left(y-1\\right)}^{2}}{4}=1,\\) we see that they are both ellipses with \\(a=3\\) and \\(b=2.\\) So they will have the same size and shape. They are different in that they do not have the same center.<\/p><span data-type=\"media\" id=\"fs-id1163872439917\" data-alt=\"The equation in the first figure is x squared upon 9 plus y squared upon 4 equals 1. Here, a is 3 and b is 2. The ellipse is graphed with center at (0, 0). The equation on the right is open parentheses x minus 3 close parentheses squared upon 9 plus open parentheses y minus 1 close parentheses squared upon 4 equals 1. Here, too, a is 3 and b is 2, but the center is (3, 1). The ellipse is shown on the same graph along with the first ellipse. The center is shown to have moved 3 units right and 1 unit up.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equation in the first figure is x squared upon 9 plus y squared upon 4 equals 1. Here, a is 3 and b is 2. The ellipse is graphed with center at (0, 0). The equation on the right is open parentheses x minus 3 close parentheses squared upon 9 plus open parentheses y minus 1 close parentheses squared upon 4 equals 1. Here, too, a is 3 and b is 2, but the center is (3, 1). The ellipse is shown on the same graph along with the first ellipse. The center is shown to have moved 3 units right and 1 unit up.\" \/><\/span><p id=\"fs-id1163872546473\"><\/p><p id=\"fs-id1163871879747\">Notice in the graph above that we could have graphed \\(\\frac{{\\left(x-3\\right)}^{2}}{9}+\\frac{{\\left(y-1\\right)}^{2}}{4}=1\\) by translations. We moved the original ellipse to the right 3 units and then up 1 unit.<\/p><span data-type=\"media\" id=\"fs-id1163872872391\" data-alt=\"This graph shows an ellipse translated from center (0, 0) to center (3, 1). The center has moved 3 units right and 1 unit up. The original ellipse has vertices at (negative 3, 0) and (3, 0) and endpoint of minor axis at (negative 2, 0) and (2, 0). The translated ellipse has vertices at (0, 1) and (6, 1) and endpoints of minor axis at (3, negative 1) and (3, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_013_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse translated from center (0, 0) to center (3, 1). The center has moved 3 units right and 1 unit up. The original ellipse has vertices at (negative 3, 0) and (3, 0) and endpoint of minor axis at (negative 2, 0) and (2, 0). The translated ellipse has vertices at (0, 1) and (6, 1) and endpoints of minor axis at (3, negative 1) and (3, 3).\" \/><\/span><p id=\"fs-id1163872436095\">In the next example we will use the translation method to graph the ellipse.<\/p><div data-type=\"example\" id=\"fs-id1163872610918\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872562232\"><div data-type=\"problem\" id=\"fs-id1163872467428\"><p id=\"fs-id1163872566832\">Graph \\(\\frac{{\\left(x+4\\right)}^{2}}{16}+\\frac{{\\left(y-6\\right)}^{2}}{9}=1\\) by translation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872514246\"><p id=\"fs-id1163872743794\">This ellipse will have the same size and shape as \\(\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1\\) whose center is \\(\\left(0,0\\right).\\) We graph this ellipse first.<span data-type=\"newline\"><br \/><\/span> <\/p><table id=\"fs-id1163872740608\" class=\"unnumbered unstyled can-break\" summary=\"The center is 0, 0). Since 16 is greater than 9, the major axis is horizontal. a squared is 16, so a is plus or minus 4. The vertices are (4, 0) and (negative 4, 0). b squared is 9 so b is plus or minus 3. The endpoints of the minor axis are (0, 3) and (0, negative 3). Graph the ellipse. The original equation is in standard form where h is minus 4 and k is 6. The center of the translated ellipse will be (negative 4, 6). We translate the graph of the first ellipse four units to the left and then up 6 units. Verify that the center is (negative 4, 6). The new ellipse has the desired equation.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The center is \\(\\left(0,0\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">Center \\(\\left(0,0\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(16&gt;9,\\) the major axis is horizontal.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2003\u2003\\({a}^{2}=16,a=\u00b14\\)<span data-type=\"newline\"><br \/><\/span>\u2003\u2003\\({b}^{2}=9,\\phantom{\\rule{0.5em}{0ex}}b=\u00b13\\)<\/td><td data-valign=\"top\" data-align=\"left\">The vertices are \\(\\left(4,0\\right),\\left(-4,0\\right).\\)<span data-type=\"newline\"><br \/><\/span>The endpoints of the minor axis are<span data-type=\"newline\"><br \/><\/span>\\(\\left(0,3\\right),\\left(0,-3\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the ellipse.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871619081\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The original equation is in standard form,<span data-type=\"newline\"><br \/><\/span>\\(\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\frac{{\\left(x-\\left(-4\\right)\\right)}^{2}}{16}+\\frac{{\\left(y-6\\right)}^{2}}{9}=1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The ellipse is centered at \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">The center is \\(\\left(-4,6\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">We translate the graph of \\(\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1\\) four<span data-type=\"newline\"><br \/><\/span>units to the left and then up 6 units.<span data-type=\"newline\"><br \/><\/span>Verify that the center is \\(\\left(-4,6\\right).\\)<span data-type=\"newline\"><br \/><\/span>The new ellipse is the ellipse whose equation<span data-type=\"newline\"><br \/><\/span>is<span data-type=\"newline\"><br \/><\/span>\\(\\frac{{\\left(x+4\\right)}^{2}}{16}+\\frac{{\\left(y-6\\right)}^{2}}{9}=1.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872534802\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163866197682\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872502349\"><div data-type=\"problem\" id=\"fs-id1163872697273\"><p id=\"fs-id1163872516184\">Graph \\(\\frac{{\\left(x-5\\right)}^{2}}{9}+\\frac{{\\left(y+4\\right)}^{2}}{4}=1\\) by translation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872388852\"><span data-type=\"media\" id=\"fs-id1163872628524\" data-alt=\"This graph shows an ellipse with center (5, negative 4), vertices (2, negative 4) and (8, negative 4) and endpoints of minor axis (5, negative 2) and (5, negative 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (5, negative 4), vertices (2, negative 4) and (8, negative 4) and endpoints of minor axis (5, negative 2) and (5, negative 6).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872537590\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872511084\"><div data-type=\"problem\" id=\"fs-id1163872569742\"><p id=\"fs-id1163872013766\">Graph \\(\\frac{{\\left(x+6\\right)}^{2}}{16}+\\frac{{\\left(y+2\\right)}^{2}}{25}=1\\) by translation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872037146\"><span data-type=\"media\" id=\"fs-id1163872426936\" data-alt=\"This graph shows an ellipse with center (negative 6, negative 2), vertices (negative 6, 3) and (negative 6, negative 7) and endpoints of minor axis (negative 10, negative 2), and (negative 2, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_309_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 6, negative 2), vertices (negative 6, 3) and (negative 6, negative 7) and endpoints of minor axis (negative 10, negative 2), and (negative 2, negative 2).\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163872666626\">When an equation has both an \\({x}^{2}\\) and a \\({y}^{2}\\) with different coefficients, we verify that it is an ellipsis by putting it in standard form. We will then be able to graph the equation.<\/p><div data-type=\"example\" id=\"fs-id1163872570378\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163872721091\"><div data-type=\"problem\" id=\"fs-id1163872836545\"><p id=\"fs-id1163871979313\">Write the equation \\({x}^{2}+4{y}^{2}-4x+24y+24=0\\) in standard form and graph.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872403290\"><p id=\"fs-id1163872837027\">We put the equation in standard form by completing the squares in both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/><\/span> <\/p><table class=\"unnumbered unstyled can-break\" summary=\"Rewrite the equation grouping the x terms and y terms. Make the coefficients of x squared and y squared equal to 1. We get open parentheses x squared minus 4 x plus close parentheses plus 4 open parentheses y squared plus 6y plus close parentheses equals minus 24. Complete the squares by adding 4 to the first term and 9 to the second term. The right side becomes minus 24 plus 4 plus 36. Write as binomial squares open parentheses x minus 2 close parentheses squared plus 4 open parentheses y plus 3 close parentheses squared equals 16. Divide both sides by 16 to get 1 on the right. The equation is in standard form with h equal to 2 and k equal to minus 3. The center is (2, negative 3). Since 16 is greater than 4 and is in the x squared term, the major axis is horizontal. a squared is 16, so a is plus or minus 4. The distance from the center to the vertices is 4. b squared is 4, so b is plus or minus 2. The distance from the center to the endpoints of the minor axis is 2. Graph the ellipse. It will have the points (2, negative 1), (2, negative 5), (6, negative 3) and (negative 2, negative 3).\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\({x}^{2}+4{y}^{2}-4x+24y+24=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite grouping the <em data-effect=\"italics\">x<\/em> terms and <em data-effect=\"italics\">y<\/em> terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872694288\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Make the coefficients of \\({x}^{2}\\) and \\({y}^{2}\\) equal 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872544205\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Complete the squares.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872479153\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write as binomial squares.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872555298\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide both sides by 16 to get 1 on the right.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872623363\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871781907\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The equation is in standard form,<span data-type=\"newline\"><br \/><\/span>\\(\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872499717\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The ellipse is centered at \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">The center is \\(\\left(2,-3\\right).\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since \\(16&gt;4\\) and 16 is in the \\({x}^{2}\\) term,<span data-type=\"newline\"><br \/><\/span>the major axis is horizontal.<span data-type=\"newline\"><br \/><\/span>\u2003\u2003\\({a}^{2}=16,a=\u00b14\\)<span data-type=\"newline\"><br \/><\/span>\u2003\u2003\\({b}^{2}=4,\\phantom{\\rule{0.5em}{0ex}}b=\u00b12\\)<\/td><td data-valign=\"bottom\" data-align=\"left\">The distance from the center to the vertices is 4.<span data-type=\"newline\"><br \/><\/span>The distance from the center to the endpoints of<span data-type=\"newline\"><br \/><\/span>the minor axis is 2.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the ellipse.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872564752\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872611551\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872012760\"><div data-type=\"problem\" id=\"fs-id1163872511507\"><p id=\"fs-id1163872841104\"><span class=\"token\">\u24d0<\/span> Write the equation \\(6{x}^{2}+4{y}^{2}+12x-32y+34=0\\) in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872464984\"><p id=\"fs-id1163872456532\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(x+1\\right)}^{2}}{6}+\\frac{{\\left(y-4\\right)}^{2}}{9}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872529490\" data-alt=\"This graph shows an ellipse with center (negative 1, 4), vertices minus (1, 1) and (negative 1, 7) and endpoints of minor axis approximately (negative 3.5, 4) and (approximately 1.5, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 1, 4), vertices minus (1, 1) and (negative 1, 7) and endpoints of minor axis approximately (negative 3.5, 4) and (approximately 1.5, 4).\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872435902\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872850405\"><div data-type=\"problem\" id=\"fs-id1163872467826\"><p id=\"fs-id1163872660171\"><span class=\"token\">\u24d0<\/span> Write the equation \\(4{x}^{2}+{y}^{2}-16x-6y+9=0\\) in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872409334\"><p id=\"fs-id1163872197218\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(x-2\\right)}^{2}}{4}+\\frac{{\\left(y-3\\right)}^{2}}{16}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872015767\" data-alt=\"This graph shows an ellipse with center (2, 3), vertices (2, negative 1) and (2, 7) and endpoints of minor axis (0, 3) and (4, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_311_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (2, 3), vertices (2, negative 1) and (2, 7) and endpoints of minor axis (0, 3) and (4, 3).\" \/><\/span><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1171792588082\"><h4 data-type=\"title\">Solve Application with Ellipses<\/h4><p id=\"fs-id1163872447476\">The orbits of the planets around the sun follow elliptical paths.<\/p><div data-type=\"example\" id=\"fs-id1163872840276\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163871987196\"><div data-type=\"problem\" id=\"fs-id1163872513963\"><p id=\"fs-id1163872529927\">Pluto (a dwarf planet) moves in an elliptical orbit around the Sun. The closest Pluto gets to the Sun is approximately 30 astronomical units (AU) and the furthest is approximately 50 AU. The Sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of Pluto.<\/p><span data-type=\"media\" id=\"fs-id1163872545123\" data-alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 40, 0) and (40, 0). The sun is shown at point (10, 0). This is 30 units from the right vertex and 50 units from the left vertex.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_016_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 40, 0) and (40, 0). The sun is shown at point (10, 0). This is 30 units from the right vertex and 50 units from the left vertex.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163868248157\"><p id=\"fs-id1163866196994\">\\(\\begin{array}{cccccc}\\begin{array}{c}\\text{We recognize this as an ellipse that is centered at the}\\hfill \\\\ \\text{origin.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{2.1em}{0ex}}\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\hfill \\\\ \\begin{array}{c}\\text{Since the major axis is horizontal and the distance from}\\hfill \\\\ \\text{the center to the vertex is 40, we know}\\phantom{\\rule{0.2em}{0ex}}a=40\\phantom{\\rule{0.2em}{0ex}}\\text{and so}\\hfill \\\\ {a}^{2}=1600.\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{1.1em}{0ex}}\\frac{{x}^{2}}{1600}+\\frac{{y}^{2}}{{b}^{2}}=1\\hfill \\\\ \\begin{array}{c}\\text{The minor axis is vertical but the end points aren\u2019t given.}\\hfill \\\\ \\text{To find}\\phantom{\\rule{0.2em}{0ex}}b\\phantom{\\rule{0.2em}{0ex}}\\text{we will use the location of the Sun. Since the}\\hfill \\\\ \\text{Sun is a focus of the ellipse at the point}\\phantom{\\rule{0.2em}{0ex}}\\left(10,0\\right),\\phantom{\\rule{0.2em}{0ex}}\\text{we know}\\hfill \\\\ c=10.\\phantom{\\rule{0.2em}{0ex}}\\text{Use this to solve for}\\phantom{\\rule{0.2em}{0ex}}{b}^{2}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{4.4em}{0ex}}\\begin{array}{c}{b}^{2}={a}^{2}-{c}^{2}\\hfill \\\\ {b}^{2}={40}^{2}-{10}^{2}\\hfill \\\\ {b}^{2}=1600-100\\hfill \\\\ {b}^{2}=1500\\hfill \\end{array}\\hfill \\\\ \\text{Substitute}\\phantom{\\rule{0.2em}{0ex}}{a}^{2}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{b}^{2}\\phantom{\\rule{0.2em}{0ex}}\\text{into the standard form of the ellipse.}\\hfill &amp; &amp; &amp; &amp; &amp; \\frac{{x}^{2}}{1600}+\\frac{{y}^{2}}{1500}=1\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872652849\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872836806\"><div data-type=\"problem\" id=\"fs-id1163872023288\"><p id=\"fs-id1163871756678\">A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately 20 AU and the furthest is approximately 30 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.<\/p><span data-type=\"media\" id=\"fs-id1163871975450\" data-alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 25, 0) and (25, 0). The sun is shown at point (5, 0). This is 20 units from the right vertex and 30 units from the left vertex.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_017_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 25, 0) and (25, 0). The sun is shown at point (5, 0). This is 20 units from the right vertex and 30 units from the left vertex.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872516912\"><p id=\"fs-id1163872092443\">\\(\\frac{{x}^{2}}{625}+\\frac{{y}^{2}}{600}=1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872436182\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163872740817\"><div data-type=\"problem\" id=\"fs-id1163872432940\"><p id=\"fs-id1163872464205\">A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately 20 AU and the furthest is approximately 50 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.<\/p><span data-type=\"media\" id=\"fs-id1163872514058\" data-alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 35, 0) and (35, 0). The sun is shown at point (15, 0). This is 20 units from the right vertex and 50 units from the left vertex.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_018_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 35, 0) and (35, 0). The sun is shown at point (15, 0). This is 20 units from the right vertex and 50 units from the left vertex.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872665959\"><p id=\"fs-id1163872713392\">\\(\\frac{{x}^{2}}{1225}+\\frac{{y}^{2}}{1000}=1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163872096564\" class=\"media-2\"><p id=\"fs-id1163872718869\">Access these online resources for additional instructions and practice with ellipses.<\/p><ul id=\"fs-id1163872503067\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37graphellipse1\">Conic Sections: Graphing Ellipses Part 1<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37graphellipse2\">Conic Sections: Graphing Ellipses Part 2<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37eqellipse\">Equation for Ellipse From Graph<\/a><\/li><\/ul><\/div><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163872697768\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1163872096217\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Ellipse:<\/strong> An <strong data-effect=\"bold\">ellipse<\/strong> is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the ellipse.<span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1163872574755\" data-alt=\"This figure shows two ellipses. In each, two points within the ellipse are labeled foci. A line drawn through the foci intersects the ellipse in two points. Each point is labeled a vertex. In The figure on the left, the segment connecting the vertices is called the major axis. A segment perpendicular to the major axis that passes through its midpoint and intersects the ellipse in two points is labeled minor axis. The major axis is longer than the minor axis. In The figure on the right, the segment through the foci, connecting the vertices is shorter and is labeled minor axis. Its midpoint is labeled center.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_020_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two ellipses. In each, two points within the ellipse are labeled foci. A line drawn through the foci intersects the ellipse in two points. Each point is labeled a vertex. In The figure on the left, the segment connecting the vertices is called the major axis. A segment perpendicular to the major axis that passes through its midpoint and intersects the ellipse in two points is labeled minor axis. The major axis is longer than the minor axis. In The figure on the right, the segment through the foci, connecting the vertices is shorter and is labeled minor axis. Its midpoint is labeled center.\" \/><\/span><span data-type=\"newline\"><br \/><\/span> If we draw a line through the foci intersects the ellipse in two points\u2014each is called a <strong data-effect=\"bold\">vertex<\/strong> of the ellipse.<span data-type=\"newline\"><br \/><\/span> The segment connecting the vertices is called the <strong data-effect=\"bold\">major axis<\/strong>.<span data-type=\"newline\"><br \/><\/span> The midpoint of the segment is called the <strong data-effect=\"bold\">center<\/strong> of the ellipse.<span data-type=\"newline\"><br \/><\/span> A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the <strong data-effect=\"bold\">minor axis<\/strong>.<\/li><li><strong data-effect=\"bold\">Standard Form of the Equation an Ellipse with Center<\/strong>\\(\\left(0,0\\right):\\) The standard form of the equation of an ellipse with center \\(\\left(0,0\\right),\\) is<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1163872624303\" class=\"unnumbered\" data-label=\"\">\\(\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\)<\/div><span data-type=\"newline\"><br \/><\/span> The <em data-effect=\"italics\">x<\/em>-intercepts are \\(\\left(a,0\\right)\\) and \\(\\left(\\text{\u2212}a,0\\right).\\)<span data-type=\"newline\"><br \/><\/span> The <em data-effect=\"italics\">y<\/em>-intercepts are \\(\\left(0,b\\right)\\) and \\(\\left(0,\\text{\u2212}b\\right).\\)<\/li><li><strong data-effect=\"bold\">How to an Ellipse with Center<\/strong>\\(\\left(0,0\\right)\\)<ol id=\"fs-id1163871923665\" type=\"1\" class=\"stepwise\"><li>Write the equation in standard form.<\/li><li>Determine whether the major axis is horizontal or vertical.<\/li><li>Find the endpoints of the major axis.<\/li><li>Find the endpoints of the minor axis<\/li><li>Sketch the ellipse.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Standard Form of the Equation an Ellipse with Center<\/strong>\\(\\left(h,k\\right):\\) The standard form of the equation of an ellipse with center \\(\\left(h,k\\right),\\) is<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1163872575490\" class=\"unnumbered\" data-label=\"\">\\(\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1\\)<\/div><span data-type=\"newline\"><br \/><\/span> When \\(a&gt;b,\\) the major axis is horizontal so the distance from the center to the vertex is <em data-effect=\"italics\">a<\/em>.<span data-type=\"newline\"><br \/><\/span> When \\(b&gt;a,\\) the major axis is vertical so the distance from the center to the vertex is <em data-effect=\"italics\">b<\/em>.<\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163872769241\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1163871876800\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1163872019091\"><strong data-effect=\"bold\">Graph an Ellipse with Center at the Origin<\/strong><\/p><p id=\"fs-id1163872014059\">In the following exercises, graph each ellipse.<\/p><div data-type=\"exercise\" id=\"fs-id1163872829148\"><div data-type=\"problem\" id=\"fs-id1163871883835\"><p id=\"fs-id1163871890082\">\\(\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872557100\"><span data-type=\"media\" id=\"fs-id1163872534088\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (2, 0) and (negative 2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (2, 0) and (negative 2, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871997735\"><div data-type=\"problem\" id=\"fs-id1163872337696\"><p id=\"fs-id1163872569434\">\\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{25}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872709615\"><div data-type=\"problem\" id=\"fs-id1163872092335\"><p id=\"fs-id1163872427624\">\\(\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{36}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872742674\"><span data-type=\"media\" id=\"fs-id1163865785818\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (5, 0) and (negative 5, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (5, 0) and (negative 5, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872110892\"><div data-type=\"problem\" id=\"fs-id1163872427472\"><p id=\"fs-id1163872103649\">\\(\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{36}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872514193\"><div data-type=\"problem\" id=\"fs-id1163871980336\"><p id=\"fs-id1163872768352\">\\(\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{16}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871926860\"><span data-type=\"media\" id=\"fs-id1163868454052\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_316_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872531832\"><div data-type=\"problem\" id=\"fs-id1163872804309\"><p id=\"fs-id1163872804311\">\\(\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{9}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872611075\"><div data-type=\"problem\" id=\"fs-id1163872560474\"><p id=\"fs-id1163872560476\">\\({x}^{2}+\\frac{{y}^{2}}{4}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872531758\"><span data-type=\"media\" id=\"fs-id1163865733298\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 2) and (0, negative 2) and endpoints of minor axis (1, 0) and (negative 1, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_318_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 2) and (0, negative 2) and endpoints of minor axis (1, 0) and (negative 1, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872768883\"><div data-type=\"problem\" id=\"fs-id1163872822384\"><p id=\"fs-id1163872542678\">\\(\\frac{{x}^{2}}{9}+{y}^{2}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163865848463\"><div data-type=\"problem\" id=\"fs-id1163865848465\"><p id=\"fs-id1163872567928\">\\(4{x}^{2}+25{y}^{2}=100\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872511469\"><span data-type=\"media\" id=\"fs-id1163872569570\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_320_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872536189\"><div data-type=\"problem\" id=\"fs-id1163872024666\"><p id=\"fs-id1163872024668\">\\(16{x}^{2}+9{y}^{2}=144\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871868040\"><div data-type=\"problem\" id=\"fs-id1163872719600\"><p id=\"fs-id1163872705504\">\\(16{x}^{2}+36{y}^{2}=576\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871935344\"><span data-type=\"media\" id=\"fs-id1163871930792\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_322_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872768062\"><div data-type=\"problem\" id=\"fs-id1163872768064\"><p id=\"fs-id1163872024700\">\\(9{x}^{2}+25{y}^{2}=225\\)<\/p><\/div><\/div><p id=\"fs-id1163872554720\"><strong data-effect=\"bold\">Find the Equation of an Ellipse with Center at the Origin<\/strong><\/p><p id=\"fs-id1163872394595\">In the following exercises, find the equation of the ellipse shown in the graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163868451657\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163868451660\"><p id=\"fs-id1163872420236\"><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872420237\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (negative 3, 0) and (3, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (negative 3, 0) and (3, 0).\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872663250\"><p id=\"fs-id1163872600785\">\\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{25}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871710691\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163871866782\"><p id=\"fs-id1163871531511\"><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163871531512\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_202_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871874103\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872521543\"><p id=\"fs-id1163872521545\"><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872110551\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 4) and (0, negative 4) and endpoints of minor axis (negative 3, 0) and (3, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_203_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 4) and (0, negative 4) and endpoints of minor axis (negative 3, 0) and (3, 0).\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872514670\"><p id=\"fs-id1163872709202\">\\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{16}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872768251\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163871992932\"><p id=\"fs-id1163871934957\"><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163871934958\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 4, 0) and (4, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_204_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 4, 0) and (4, 0).\" \/><\/span><\/div><\/div><p id=\"fs-id1163872015114\"><strong data-effect=\"bold\">Graph an Ellipse with Center Not at the Origin<\/strong><\/p><p>In the following exercises, graph each ellipse.<\/p><div data-type=\"exercise\" id=\"fs-id1163872418395\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872418397\"><p id=\"fs-id1163871987398\">\\(\\frac{{\\left(x+1\\right)}^{2}}{4}+\\frac{{\\left(y+6\\right)}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871994520\"><span data-type=\"media\" id=\"fs-id1163872096387\" data-alt=\"This graph shows an ellipse with center (negative 1, negative 6, vertices (negative 1, negative 1) and (negative 1, negative 11) and endpoints of minor axis (negative 3, negative 6) and (1, negative 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_324_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 1, negative 6, vertices (negative 1, negative 1) and (negative 1, negative 11) and endpoints of minor axis (negative 3, negative 6) and (1, negative 6).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872570186\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872533709\"><p id=\"fs-id1163872533711\">\\(\\frac{{\\left(x-3\\right)}^{2}}{25}+\\frac{{\\left(y+2\\right)}^{2}}{9}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872842208\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872612895\"><p id=\"fs-id1163872612897\">\\(\\frac{{\\left(x+4\\right)}^{2}}{4}+\\frac{{\\left(y-2\\right)}^{2}}{9}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872302554\"><span data-type=\"media\" id=\"fs-id1163872108528\" data-alt=\"This graph shows an ellipse with center (negative 4, 2, vertices (negative 4, 5) and (negative 4, negative 1) and endpoints of minor axis (3, 1) and (negative 6, 2) and (negative 2, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_326_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 4, 2, vertices (negative 4, 5) and (negative 4, negative 1) and endpoints of minor axis (3, 1) and (negative 6, 2) and (negative 2, 2).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872448776\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872720576\"><p id=\"fs-id1163872720578\">\\(\\frac{{\\left(x-4\\right)}^{2}}{9}+\\frac{{\\left(y-1\\right)}^{2}}{16}=1\\)<\/p><\/div><\/div><p id=\"fs-id1163871932647\">In the following exercises, graph each equation by translation.<\/p><div data-type=\"exercise\" id=\"fs-id1163871783180\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872022197\"><p id=\"fs-id1163872022200\">\\(\\frac{{\\left(x-3\\right)}^{2}}{4}+\\frac{{\\left(y-7\\right)}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872708891\"><span data-type=\"media\" id=\"fs-id1163872799939\" data-alt=\"This graph shows an ellipse with center (3, 7), vertices (3, 2) and (3, 12), and endpoints of minor axis (1, 7) and (5, 7).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_328_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (3, 7), vertices (3, 2) and (3, 12), and endpoints of minor axis (1, 7) and (5, 7).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871882005\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872403279\"><p id=\"fs-id1163872403281\">\\(\\frac{{\\left(x+6\\right)}^{2}}{16}+\\frac{{\\left(y+5\\right)}^{2}}{4}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871927127\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872453024\"><p id=\"fs-id1163871985605\">\\(\\frac{{\\left(x-5\\right)}^{2}}{9}+\\frac{{\\left(y+4\\right)}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872630902\"><span data-type=\"media\" id=\"fs-id1163872444389\" data-alt=\"This graph shows an ellipse with center (5, negative 4), vertices (5, 1) and (5, negative 9) and endpoints of minor axis (2, negative 4) and (8, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_330_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (5, negative 4), vertices (5, 1) and (5, negative 9) and endpoints of minor axis (2, negative 4) and (8, negative 4).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872510998\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872014761\"><p id=\"fs-id1163871378181\">\\(\\frac{{\\left(x+5\\right)}^{2}}{36}+\\frac{{\\left(y-3\\right)}^{2}}{16}=1\\)<\/p><\/div><\/div><p id=\"fs-id1163872833573\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163872693587\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872573326\"><p id=\"fs-id1163872553270\">\\(25{x}^{2}+9{y}^{2}-100x-54y-44=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872660997\"><p id=\"fs-id1163872034704\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(x-2\\right)}^{2}}{9}+\\frac{{\\left(y-3\\right)}^{2}}{25}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163871934818\" data-alt=\"This graph shows an ellipse with center (2, 3), vertices (2, negative 2) and (2, 8) and endpoints of minor axis (negative 1, 3) and (5, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_332_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (2, 3), vertices (2, negative 2) and (2, 8) and endpoints of minor axis (negative 1, 3) and (5, 3).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872841555\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163871933662\"><p id=\"fs-id1163872743279\">\\(4{x}^{2}+25{y}^{2}+8x+100y+4=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871704288\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872397377\"><p id=\"fs-id1163872639909\">\\(4{x}^{2}+25{y}^{2}-24x-64=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163868452715\"><p id=\"fs-id1163872446383\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{y}^{2}}{4}+\\frac{{\\left(x-3\\right)}^{2}}{25}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163872093634\" data-alt=\"This graph shows an ellipse with center (3, 0), vertices (negative 2, 0) and (8, 0) and endpoints of minor axis (3, 2) and (3, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_334_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (3, 0), vertices (negative 2, 0) and (8, 0) and endpoints of minor axis (3, 2) and (3, negative 2).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872515061\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872515063\"><p id=\"fs-id1163872095366\">\\(9{x}^{2}+4{y}^{2}+56y+160=0\\)<\/p><\/div><\/div><p id=\"fs-id1163872469152\">In the following exercises, graph the equation.<\/p><div data-type=\"exercise\" id=\"fs-id1163872400116\"><div data-type=\"problem\" id=\"fs-id1163872400118\"><p id=\"fs-id1163872693934\">\\(x=-2{\\left(y-1\\right)}^{2}+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872093400\"><span data-type=\"media\" id=\"fs-id1163871349264\" data-alt=\"This graph shows a parabola with vertex (2, 1) and y intercepts (0, 0) and (2, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_336_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola with vertex (2, 1) and y intercepts (0, 0) and (2, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163866047685\"><div data-type=\"problem\" id=\"fs-id1163872467848\"><p id=\"fs-id1163872559529\">\\({x}^{2}+{y}^{2}=49\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872600142\"><div data-type=\"problem\" id=\"fs-id1163872600144\"><p id=\"fs-id1163872833517\">\\({\\left(x+5\\right)}^{2}+{\\left(y+2\\right)}^{2}=4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872243575\"><span data-type=\"media\" id=\"fs-id1163872742897\" data-alt=\"This graph shows a circle with center (negative 5, negative 2) and a radius of 2 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_338_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a circle with center (negative 5, negative 2) and a radius of 2 units.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872664239\"><div data-type=\"problem\" id=\"fs-id1163872646173\"><p id=\"fs-id1163872666370\">\\(y=\\text{\u2212}{x}^{2}+8x-15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871756540\"><div data-type=\"problem\" id=\"fs-id1163872034683\"><p id=\"fs-id1163872576509\">\\(\\frac{{\\left(x+3\\right)}^{2}}{16}+\\frac{{\\left(y+1\\right)}^{2}}{4}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871567468\"><span data-type=\"media\" id=\"fs-id1163865730465\" data-alt=\"This graph shows an ellipse with center (negative 3, negative 1), vertices (1, negative 1) and (negative 7, negative 1) and endpoints of minor axis (negative 3, 1) and (negative 3, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_340_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 3, negative 1), vertices (1, negative 1) and (negative 7, negative 1) and endpoints of minor axis (negative 3, 1) and (negative 3, negative 3).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872557744\"><div data-type=\"problem\" id=\"fs-id1163872630243\"><p id=\"fs-id1163866226794\">\\({\\left(x-2\\right)}^{2}+{\\left(y-3\\right)}^{2}=9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872629661\"><div data-type=\"problem\" id=\"fs-id1163872629663\"><p id=\"fs-id1163871930849\">\\(\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{36}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872642609\"><span data-type=\"media\" id=\"fs-id1163866198996\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 5, 0) and (5, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_342_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 5, 0) and (5, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871866286\"><div data-type=\"problem\" id=\"fs-id1163871866288\"><p id=\"fs-id1163872462579\">\\(x=4{\\left(y+1\\right)}^{2}-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871908392\"><div data-type=\"problem\" id=\"fs-id1163871908394\"><p id=\"fs-id1163871859292\">\\({x}^{2}+{y}^{2}=64\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163868434169\"><span data-type=\"media\" id=\"fs-id1163872532047\" data-alt=\"This graph shows circle with center (0, 0) and with radius 8 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_344_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (0, 0) and with radius 8 units.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871870368\"><div data-type=\"problem\" id=\"fs-id1163871870370\"><p id=\"fs-id1163872799994\">\\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{25}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872560516\"><div data-type=\"problem\" id=\"fs-id1163872840603\"><p id=\"fs-id1163872840605\">\\(y=6{x}^{2}+2x-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871731254\"><span data-type=\"media\" id=\"fs-id1163872033812\" data-alt=\"This graph shows upward opening parabola. Its vertex has an x value of slightly less than 0 and a y value of slightly less than minus 1. A point on it is approximately at (negative 1, 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_346_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows upward opening parabola. Its vertex has an x value of slightly less than 0 and a y value of slightly less than minus 1. A point on it is approximately at (negative 1, 3).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872940076\"><div data-type=\"problem\" id=\"fs-id1163872729685\"><p id=\"fs-id1163872729687\">\\(\\frac{{\\left(x-2\\right)}^{2}}{9}+\\frac{{\\left(y+3\\right)}^{2}}{25}=1\\)<\/p><\/div><\/div><p id=\"fs-id1163872472137\"><strong data-effect=\"bold\">Solve Application with Ellipses<\/strong><\/p><div data-type=\"exercise\" id=\"fs-id1163871987462\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163871987464\"><p id=\"fs-id1163872562521\">A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately 10 AU and the furthest is approximately 30 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.<\/p><span data-type=\"media\" id=\"fs-id1163872562525\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 20, 0) and (20, 0). The sun is shown at point (10, 0), which is 30 units from the left vertex and 10 units from the right vertex.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_205_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 20, 0) and (20, 0). The sun is shown at point (10, 0), which is 30 units from the left vertex and 10 units from the right vertex.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872548932\"><p id=\"fs-id1163872105286\">\\(\\frac{{x}^{2}}{400}+\\frac{{y}^{2}}{300}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872435929\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163872435931\"><p id=\"fs-id1163872404828\">A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately 10 AU and the furthest is approximately 70 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.<\/p><span data-type=\"media\" id=\"fs-id1163872404831\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 40, 0) and (40, 0). The sun is shown at point (30, 0), which is 70 units from the left vertex and 10 units from the right vertex.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_206_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 40, 0) and (40, 0). The sun is shown at point (30, 0), which is 70 units from the left vertex and 10 units from the right vertex.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163871895792\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163871895794\"><p id=\"fs-id1163872556219\">A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 15 AU and the furthest is approximately 85 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.<\/p><span data-type=\"media\" id=\"fs-id1163872556221\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 50, 0) and (50, 0). The sun is shown at point (35, 0), which is 85 units from the left vertex and 15 units from the right vertex.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_207_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 50, 0) and (50, 0). The sun is shown at point (35, 0), which is 85 units from the left vertex and 15 units from the right vertex.\" \/><\/span><\/div><div data-type=\"solution\" id=\"fs-id1163872561506\"><p id=\"fs-id1163872561508\">\\(\\frac{{x}^{2}}{2500}+\\frac{{y}^{2}}{1275}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872748963\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163871704679\"><p id=\"fs-id1163871704681\">A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 15 AU and the furthest is approximately 95 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.<\/p><span data-type=\"media\" id=\"fs-id1163872599043\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 55, 0) and (55, 0). The sun is shown at point (40, 0), which is 95 units from the left vertex and 15 units from the right vertex.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_208_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 55, 0) and (55, 0). The sun is shown at point (40, 0), which is 95 units from the left vertex and 15 units from the right vertex.\" \/><\/span><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1163863474708\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1163871993197\"><div data-type=\"problem\" id=\"fs-id1163871993199\"><p id=\"fs-id1163871977038\">In your own words, define an ellipse and write the equation of an ellipse centered at the origin in standard form. Draw a sketch of the ellipse labeling the center, vertices and major and minor axes.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163871977040\"><p id=\"fs-id1163872800713\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872564055\"><div data-type=\"problem\" id=\"fs-id1163872011994\"><p id=\"fs-id1163872011996\">Explain in your own words how to get the axes from the equation in standard form.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872713558\"><div data-type=\"problem\" id=\"fs-id1163872713560\"><p id=\"fs-id1163872546052\">Compare and contrast the graphs of the equations \\(\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1\\) and \\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{4}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163872713406\"><p id=\"fs-id1163872713408\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163872020384\"><div data-type=\"problem\" id=\"fs-id1163872435908\"><p id=\"fs-id1163872435910\">Explain in your own words, the difference between a vertex and a focus of the ellipse.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163872095843\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1163872402231\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1163872548678\" data-alt=\"This table has 4 columns 4 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don&#x2019;t get it. The first columns has the following statements: graph an ellipse with center at the origin, find the equation of an ellipse with center at the origin, graph an ellipse with center not at the origin, solve applications with ellipses. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_209_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns 4 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don&#x2019;t get it. The first columns has the following statements: graph an ellipse with center at the origin, find the equation of an ellipse with center at the origin, graph an ellipse with center not at the origin, solve applications with ellipses. The remaining columns are blank.\" \/><\/span><p id=\"fs-id1163871568158\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1163871993300\"><dt>ellipse<\/dt><dd id=\"fs-id1163872747198\">An ellipse is all points in a plane where the sum of the distances from two fixed points is constant.<\/dd><\/dl><\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Graph an ellipse with center at the origin<\/li>\n<li>Find the equation of an ellipse with center at the origin<\/li>\n<li>Graph an ellipse with center not at the origin<\/li>\n<li>Solve application with ellipses<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872103152\" class=\"be-prepared\">\n<p id=\"fs-id1163872387647\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1163872108629\" type=\"1\">\n<li>Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38667f5b7cf18628070a43238051d263_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/> using transformations.<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/9054900c-f191-40ed-8ceb-e4f051bbbf2b#fs-id1169148912189\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Complete the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b2efe612ce3442ce5b5d31ed285e9a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: 0px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/3e7f365d-4885-4b7a-bb2a-4358cbc00d2e#fs-id1167829894368\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Write in standard form. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf8f3f990615a96b3ff0b64432399ba3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#43;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/9054900c-f191-40ed-8ceb-e4f051bbbf2b#fs-id1169149374763\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872520186\">\n<h3 data-type=\"title\">Graph an Ellipse with Center at the Origin<\/h3>\n<p id=\"fs-id1163872416455\">The next conic section we will look at is an <span data-type=\"term\">ellipse<\/span>. We define an ellipse as all points in a plane where the sum of the distances from two fixed points is constant. Each of the given points is called a <strong data-effect=\"bold\">focus<\/strong> of the ellipse.<\/p>\n<div data-type=\"note\" id=\"fs-id1163872730452\">\n<div data-type=\"title\">Ellipse<\/div>\n<p id=\"fs-id1163872721606\">An <strong data-effect=\"bold\">ellipse<\/strong> is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the ellipse.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872515324\" data-alt=\"This figure shows a double cone intersected by a plane to form an ellipse.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a double cone intersected by a plane to form an ellipse.\" \/><\/span><\/div>\n<p id=\"fs-id1163872009594\">We can draw an ellipse by taking some fixed length of flexible string and attaching the ends to two thumbtacks. We use a pen to pull the string taut and rotate it around the two thumbtacks. The figure that results is an ellipse.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163871857451\" data-alt=\"This figure shows a pen attached to two strings, the other ends of which are attached to two thumbtacks. The strings are pulled taut and the pen is rotated to draw an ellipse. The thumbtacks are labeled F subscript 1 and F subscript 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a pen attached to two strings, the other ends of which are attached to two thumbtacks. The strings are pulled taut and the pen is rotated to draw an ellipse. The thumbtacks are labeled F subscript 1 and F subscript 2.\" \/><\/span><\/p>\n<p id=\"fs-id1163868232249\">A line drawn through the foci intersect the ellipse in two points. Each point is called a <strong data-effect=\"bold\">vertex<\/strong> of the ellipse. The segment connecting the vertices is called the <strong data-effect=\"bold\">major axis<\/strong>. The midpoint of the segment is called the <strong data-effect=\"bold\">center<\/strong> of the ellipse. A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the <strong data-effect=\"bold\">minor axis<\/strong>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163866197380\" data-alt=\"This figure shows two ellipses. In each, two points within the ellipse are labeled foci. A line drawn through the foci intersects the ellipse in two points. Each point is labeled a vertex. In The figure on the left, the segment connecting the vertices is called the major axis. A segment perpendicular to the major axis that passes through its midpoint and intersects the ellipse in two points is labeled minor axis. The major axis is longer than the minor axis. In The figure on the right, the segment through the foci, connecting the vertices is shorter and is labeled minor axis. Its midpoint is labeled center.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two ellipses. In each, two points within the ellipse are labeled foci. A line drawn through the foci intersects the ellipse in two points. Each point is labeled a vertex. In The figure on the left, the segment connecting the vertices is called the major axis. A segment perpendicular to the major axis that passes through its midpoint and intersects the ellipse in two points is labeled minor axis. The major axis is longer than the minor axis. In The figure on the right, the segment through the foci, connecting the vertices is shorter and is labeled minor axis. Its midpoint is labeled center.\" \/><\/span><\/p>\n<p id=\"fs-id1163871867164\">We mentioned earlier that our goal is to connect the geometry of a conic with algebra. Placing the ellipse on a rectangular coordinate system gives us that opportunity. In the figure, we placed the ellipse so the foci <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-648c27f35a5afa4689612967894ad52a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#99;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/> are on the <em data-effect=\"italics\">x<\/em>-axis and the center is the origin.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872705598\" data-alt=\"The figure on the left shows an ellipse with its center at the origin of the coordinate axes and its foci at points minus (c, 0) and (c, 0). A segment connects (negative c, 0) to a point (x, y) on the ellipse. The segment is labeled d subscript 1. Another segment, labeled d subscript 2 connects (c, 0) to (x, y). The figure on the right shows an ellipse with center at the origin, foci (negative c, 0) and (c, 0) and vertices (negative a, 0) and (a, 0). The point where the ellipse intersects the y axis is labeled (0, b). The segments connecting (0, 0) to (c, 0), (c, 0) to (0, b) and (0, b) to (0, 0) form a tight angled triangle with sides c, a and b respectively. The equation is a squared equals b squared plus c squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure on the left shows an ellipse with its center at the origin of the coordinate axes and its foci at points minus (c, 0) and (c, 0). A segment connects (negative c, 0) to a point (x, y) on the ellipse. The segment is labeled d subscript 1. Another segment, labeled d subscript 2 connects (c, 0) to (x, y). The figure on the right shows an ellipse with center at the origin, foci (negative c, 0) and (c, 0) and vertices (negative a, 0) and (a, 0). The point where the ellipse intersects the y axis is labeled (0, b). The segments connecting (0, 0) to (c, 0), (c, 0) to (0, b) and (0, b) to (0, 0) form a tight angled triangle with sides c, a and b respectively. The equation is a squared equals b squared plus c squared.\" \/><\/span><\/p>\n<p id=\"fs-id1163871890058\">The definition states the sum of the distance from the foci to a point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> is constant. So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cda187a78938b339b1f6ef31b344dd52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#100;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"55\" style=\"vertical-align: -4px;\" \/> is a constant that we will call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02baaffe4ddb6a44400eb7ba175e566c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> so, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebd480c3d4944df2d8ae8fb9bd7e8666_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#61;&#50;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -4px;\" \/> We will use the distance formula to lead us to an algebraic formula for an ellipse.<\/p>\n<div data-type=\"equation\" id=\"fs-id1163872419619\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3329ffea9983c2a1dae7a11349fce884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#32;&#102;&#111;&#114;&#109;&#117;&#108;&#97;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#102;&#116;&#101;&#114;&#32;&#101;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#105;&#110;&#103;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#115;&#32;&#97;&#110;&#100;&#32;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#105;&#110;&#103;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#101;&#32;&#103;&#101;&#116;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#44;&#32;&#119;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#44;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#97;&#110;&#32;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#32;&#99;&#101;&#110;&#116;&#101;&#114;&#101;&#100;&#32;&#97;&#116;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#105;&#103;&#105;&#110;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#32;&#105;&#115;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"285\" width=\"868\" style=\"vertical-align: -146px;\" \/><\/div>\n<p id=\"fs-id1163871567318\">To graph the ellipse, it will be helpful to know the intercepts. We will find the <em data-effect=\"italics\">x<\/em>-intercepts and <em data-effect=\"italics\">y<\/em>-intercepts using the formula.<\/p>\n<div data-type=\"equation\" id=\"fs-id1163872646544\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0cbb033533df5b38597dac21511b67f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&plusmn;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&plusmn;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#32;&#97;&#114;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#32;&#97;&#114;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"170\" width=\"632\" style=\"vertical-align: -81px;\" \/><\/div>\n<div data-type=\"note\" id=\"fs-id1163871935632\">\n<div data-type=\"title\">Standard Form of the Equation an Ellipse with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14389c22b5058167cc4367ebe83dfbf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1163872096452\">The standard form of the equation of an ellipse with center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8cbcb1fb27edd3e5c2aa2c764d762045_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> is<\/p>\n<div data-type=\"equation\" id=\"fs-id1163872630602\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4100e34708a781839b2e089b9992ca73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"fs-id1163868452315\">The <em data-effect=\"italics\">x<\/em>-intercepts are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-914e9523c72b2e0b3d6961c80651140f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1163872541971\">The <em data-effect=\"italics\">y<\/em>-intercepts are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05dbabd42fe84ba97e673be0628c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42fd44aa7d2ea598ea3ef7b69187f6b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872938165\" data-alt=\"Two figures show ellipses with their centers on the origin of the coordinate axes. They intersect the x axis at points (negative a, 0) and (a, 0) and the y axis at points (0, b) and (0, negative b). In the figure on the left the major axis of the ellipse is along the x axis and in the figure on the right, it is along the y axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_005_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two figures show ellipses with their centers on the origin of the coordinate axes. They intersect the x axis at points (negative a, 0) and (a, 0) and the y axis at points (0, b) and (0, negative b). In the figure on the left the major axis of the ellipse is along the x axis and in the figure on the right, it is along the y axis.\" \/><\/span><\/div>\n<p id=\"fs-id1163871911233\">Notice that when the major axis is horizontal, the value of <em data-effect=\"italics\">a<\/em> will be greater than the value of <em data-effect=\"italics\">b<\/em> and when the major axis is vertical, the value of <em data-effect=\"italics\">b<\/em> will be greater than the value of <em data-effect=\"italics\">a<\/em>. We will use this information to graph an <span data-type=\"term\" class=\"no-emphasis\">ellipse<\/span> that is centered at the origin.<\/p>\n<table id=\"fs-id1163872513141\" summary=\".\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Ellipse with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4100e34708a781839b2e089b9992ca73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b84b1480aef6484626cffaeccec0b9fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f32782d6aab14f49821cc4f76148e931_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#62;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Major axis<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">on the <em data-effect=\"italics\">x<\/em>&#8211; axis.<\/td>\n<td data-valign=\"middle\" data-align=\"center\">on the <em data-effect=\"italics\">y<\/em>-axis.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td>\n<td colspan=\"2\" data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9959f9bc5b535a53482e1f95d8e47160_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td>\n<td colspan=\"2\" data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6556c12066af78ce791ca65908b97ce5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05dbabd42fe84ba97e673be0628c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" id=\"fs-id1163872731080\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Graph an Ellipse with Center (0, 0)<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163868248185\">\n<div data-type=\"problem\" id=\"fs-id1163872022977\">\n<p id=\"fs-id1163872468441\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88f738f32d869f222737900f7ae6f4de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872538228\"><span data-type=\"media\" id=\"fs-id1163872427329\" data-alt=\"Step 1. Write the equation in standard form. It is in standard form x squared upon 6 plus y squared upon 9 equals 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1. Write the equation in standard form. It is in standard form x squared upon 6 plus y squared upon 9 equals 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163872399906\" data-alt=\"Step 2. Determine whether the major axis is horizontal or vertical. Since 9 is greater than 4 and 9 is in the y squared term, the major axis is vertical.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. Determine whether the major axis is horizontal or vertical. Since 9 is greater than 4 and 9 is in the y squared term, the major axis is vertical.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163872642709\" data-alt=\"Step 3. Find the endpoints of the major axis. The endpoints will be the y-intercepts. Since b squared is 9, b is plus or minus 3. The endpoints of the major axis are (0, 3) and (0, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. Find the endpoints of the major axis. The endpoints will be the y-intercepts. Since b squared is 9, b is plus or minus 3. The endpoints of the major axis are (0, 3) and (0, negative 3).\" \/><\/span><span data-type=\"media\" id=\"fs-id1163871870410\" data-alt=\"Step 4. Find the endpoints of the minor axis. The endpoints will be the x-intercepts. Since a squared is 4, a is plus or minus 2. The endpoints of the minor axis are (2, 0) and (negative 2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Find the endpoints of the minor axis. The endpoints will be the x-intercepts. Since a squared is 4, a is plus or minus 2. The endpoints of the minor axis are (2, 0) and (negative 2, 0).\" \/><\/span><span data-type=\"media\" id=\"fs-id1163872842289\" data-alt=\"Step 5. Sketch the ellipse using the x and y intercepts. The graph shows an ellipse with center at (0, 0) and foci at (0, 3), (0, negative 3), (negative 2, 0), and (2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5. Sketch the ellipse using the x and y intercepts. The graph shows an ellipse with center at (0, 0) and foci at (0, 3), (0, negative 3), (negative 2, 0), and (2, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872473001\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872449228\">\n<div data-type=\"problem\" id=\"fs-id1163872654950\">\n<p id=\"fs-id1163872435556\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73ff99b2ebf286a9f686938002dad8fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"93\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872667846\"><span data-type=\"media\" id=\"fs-id1163872543638\" data-alt=\"This graph shows an ellipse with x intercepts (negative 2, 0) and (2, 0) and y intercepts (0, 4) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 2, 0) and (2, 0) and y intercepts (0, 4) and (0, negative 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872560939\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872515144\">\n<div data-type=\"problem\" id=\"fs-id1163871870306\">\n<p id=\"fs-id1163872727725\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-358919906c97eb82f30084398984c93d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"93\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872470405\"><span data-type=\"media\" id=\"fs-id1163871979084\" data-alt=\"This graph shows an ellipse with x intercepts (negative 3, 0) and (3, 0) and y intercepts (0, 4) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 3, 0) and (3, 0) and y intercepts (0, 4) and (0, negative 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872513225\">We summarize the steps for reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1163872664715\" class=\"howto\">\n<div data-type=\"title\">How to Graph an Ellipse with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/div>\n<ol id=\"fs-id1163872554151\" type=\"1\" class=\"stepwise\">\n<li>Write the equation in standard form.<\/li>\n<li>Determine whether the major axis is horizontal or vertical.<\/li>\n<li>Find the endpoints of the major axis.<\/li>\n<li>Find the endpoints of the minor axis<\/li>\n<li>Sketch the ellipse.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1163871890228\">Sometimes our equation will first need to be put in standard form.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872427815\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872611858\">\n<div data-type=\"problem\" id=\"fs-id1163872539321\">\n<p id=\"fs-id1163872391526\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-528ee7ade88eb3b36e0e8fa6b54bfdf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163868452816\">\n<table id=\"fs-id1163872012967\" class=\"unnumbered unstyled can-break\" summary=\"The equation is x squared plus 4 y squared equals 16. We recognize this as the equation of an ellipse since both the x and y terms are squared and have different coefficients. To get the equation in standard form, divide both sides by 16 so that the right side of the equation is equal to 1. Simplify to get x squared upon 16 plus y squared upon 4 equals 1. The equation is in standard form. The ellipse is centered at the origin, (0, 0). Since 16 is greater than 4 and 16 is in the x squared term, the major axis is horizontal. a squared is 16, so a is plus or minus 4. The vertices are (4, 0) and (negative 4, 0). b squared is 4, so b is plus or minus 2. The endpoints of the minor axis are (0, 2) and (0, negative 2). Sketch the ellipse.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We recognize this as the equation of an<span data-type=\"newline\"><br \/><\/span>ellipse since both the <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> terms are<span data-type=\"newline\"><br \/><\/span>squared and have different coefficients.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-060ee24469dfb0418389077f60d57263_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To get the equation in standard form, divide<span data-type=\"newline\"><br \/><\/span>both sides by 16 so that the equation is equal<span data-type=\"newline\"><br \/><\/span>to 1.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-859ec5f1e76cf33b525c8b90c3449148_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"102\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97ee3cf77a81b4e85b238ec768494f6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"87\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The equation is in standard form.<span data-type=\"newline\"><br \/><\/span>The ellipse is centered at the origin.<\/td>\n<td data-valign=\"top\" data-align=\"center\">The center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee90a82eb157fe207819bba51733883b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: -1px;\" \/> and 16 is in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term,<span data-type=\"newline\"><br \/><\/span>the major axis is horizontal.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d48b740446b4dfb00b2b8fffa30e8a2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#44;&#97;&#61;&plusmn;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-435d37a36173dd14de3261cbf5de7409_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;&plusmn;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">The vertices are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e4cd56d89d31e40a781c004db11511a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>The endpoints of the minor axis are<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de777c6a9633ee1ebdfc63cb7b863881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the parabola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872562489\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872840418\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872388297\">\n<div data-type=\"problem\" id=\"fs-id1163872557543\">\n<p id=\"fs-id1163872425484\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-014eb07686acbb1a8d14dc29a1db4673_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872446494\"><span data-type=\"media\" id=\"fs-id1163872542325\" data-alt=\"This graph shows an ellipse with x intercepts (negative 4, 0) and (4, 0) and y intercepts (0, 3) and (0, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 4, 0) and (4, 0) and y intercepts (0, 3) and (0, negative 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872691837\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872437330\">\n<div data-type=\"problem\" id=\"fs-id1163872016439\">\n<p id=\"fs-id1163872512504\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fab1f201b6ad50eb41ff838a3e2ee8c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#48;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872694456\"><span data-type=\"media\" id=\"fs-id1163871860728\" data-alt=\"This graph shows an ellipse with x intercepts (negative 5, 0) and (5, 0) and y intercepts (0, 4) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 5, 0) and (5, 0) and y intercepts (0, 4) and (0, negative 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872654447\">\n<h3 data-type=\"title\">Find the Equation of an Ellipse with Center at the Origin<\/h3>\n<p id=\"fs-id1163871865681\">If we are given the graph of an <span data-type=\"term\" class=\"no-emphasis\">ellipse<\/span>, we can find the equation of the ellipse.<\/p>\n<div data-type=\"example\" id=\"fs-id1163868321690\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872829077\">\n<div data-type=\"problem\" id=\"fs-id1163872728911\">\n<p id=\"fs-id1163872714191\">Find the equation of the ellipse shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872544767\" data-alt=\"This graph shows an ellipse with x intercepts (negative 4, 0) and (4, 0) and y intercepts (0, 3) and (0, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_008_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 4, 0) and (4, 0) and y intercepts (0, 3) and (0, negative 3).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872446354\">\n<p id=\"fs-id1163872420512\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d94ead394e07094eceec8ab553592216_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#114;&#101;&#99;&#111;&#103;&#110;&#105;&#122;&#101;&#32;&#116;&#104;&#105;&#115;&#32;&#97;&#115;&#32;&#97;&#110;&#32;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#32;&#116;&#104;&#97;&#116;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#101;&#110;&#116;&#101;&#114;&#101;&#100;&#32;&#97;&#116;&#32;&#116;&#104;&#101;&#32;&#111;&#114;&#105;&#103;&#105;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#109;&#97;&#106;&#111;&#114;&#32;&#97;&#120;&#105;&#115;&#32;&#105;&#115;&#32;&#104;&#111;&#114;&#105;&#122;&#111;&#110;&#116;&#97;&#108;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#32;&#102;&#114;&#111;&#109;&#32;&#116;&#104;&#101;&#32;&#99;&#101;&#110;&#116;&#101;&#114;&#32;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#118;&#101;&#114;&#116;&#101;&#120;&#32;&#105;&#115;&#32;&#52;&#44;&#32;&#119;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#110;&#111;&#119;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#61;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#109;&#105;&#110;&#111;&#114;&#32;&#97;&#120;&#105;&#115;&#32;&#105;&#115;&#32;&#118;&#101;&#114;&#116;&#105;&#99;&#97;&#108;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#114;&#111;&#109;&#32;&#116;&#104;&#101;&#32;&#99;&#101;&#110;&#116;&#101;&#114;&#32;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#32;&#105;&#115;&#32;&#51;&#44;&#32;&#119;&#101;&#32;&#107;&#110;&#111;&#119;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#98;&#61;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"537\" style=\"vertical-align: -77px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163871618969\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872750117\">\n<div data-type=\"problem\" id=\"fs-id1163872836610\">\n<p id=\"fs-id1163872511100\">Find the equation of the ellipse shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872565747\" data-alt=\"This graph shows an ellipse with x intercepts (negative 2, 0) and (2, 0) and y intercepts (0, 5) and (0, negative 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_009_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 2, 0) and (2, 0) and y intercepts (0, 5) and (0, negative 5).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872556813\">\n<p id=\"fs-id1163872638056\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24c6a0e61582bcbdd04b9fed7f35cd69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163871930788\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872547351\">\n<div data-type=\"problem\" id=\"fs-id1163868453510\">\n<p id=\"fs-id1163872607559\">Find the equation of the ellipse shown.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872690626\" data-alt=\"This graph shows an ellipse with x intercepts (negative 3, 0) and (3, 0) and y intercepts (0, 2) and (0, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_010_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with x intercepts (negative 3, 0) and (3, 0) and y intercepts (0, 2) and (0, negative 2).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163871980514\">\n<p id=\"fs-id1163872689651\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5980e076b60afda408a49959150c832d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163872533788\">\n<h3 data-type=\"title\">Graph an Ellipse with Center Not at the Origin<\/h3>\n<p id=\"fs-id1163872643895\">The ellipses we have looked at so far have all been centered at the origin. We will now look at ellipses whose center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1163872545176\">The equation is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdc35d2c510f0d6e68352ba88eea5b4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"146\" style=\"vertical-align: -7px;\" \/> and when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b069cd4e45edfcf03bedc4812930311_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#98;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/> the major axis is horizontal so the distance from the center to the vertex is <em data-effect=\"italics\">a<\/em>. When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-241db334cff6d4114e26d1794f470af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#62;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> the major axis is vertical so the distance from the center to the vertex is <em data-effect=\"italics\">b<\/em>.<\/p>\n<div data-type=\"note\" id=\"fs-id1163872729923\">\n<div data-type=\"title\">Standard Form of the Equation an Ellipse with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> <\/div>\n<p id=\"fs-id1163872447153\">The standard form of the equation of an ellipse with center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> is<\/p>\n<div data-type=\"equation\" id=\"fs-id1163872564491\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdc35d2c510f0d6e68352ba88eea5b4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"146\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"fs-id1163871867293\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b069cd4e45edfcf03bedc4812930311_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#98;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/> the major axis is horizontal so the distance from the center to the vertex is <em data-effect=\"italics\">a<\/em>.<\/p>\n<p id=\"fs-id1163871881869\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-241db334cff6d4114e26d1794f470af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#62;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> the major axis is vertical so the distance from the center to the vertex is <em data-effect=\"italics\">b<\/em>.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1163872427883\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872700016\">\n<div data-type=\"problem\" id=\"fs-id1163871999656\">\n<p id=\"fs-id1163872463704\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2c5bab2ecba509d72c01dd207376667_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872017878\">\n<table id=\"fs-id1163872729668\" class=\"unnumbered unstyled\" summary=\"The equation is open parentheses x minus 3 close parentheses squared upon 9 plus open parentheses y minus 1 close parentheses squared upon 4 equals 1. This is in standard form. The ellipse is centered at (h, k), which is (3, 1). Since 9 is greater than 4 and is in the x squared term, the major axis is horizontal. a squared is 9, so a is plus or minus 3. The distance from the center to the vertices is 3. b squared is 4, so b is plus or minus 2. The distance from the center to the endpoints of the minor axis is 2. Sketch the ellipse with point 3, 3, point 3, negative 3, point 6, 1) and (point 0, 1).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The equation is in standard form,<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-155bd1abd30882967a1a7f5db788ecb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"151\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcc7e77e53ee242e07928690a4aecdb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The ellipse is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">The center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c867e408365ba7dbd665695abcf20b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee5844693262d6e8ab4400cf2b861dc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/> and 9 is in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term,<span data-type=\"newline\"><br \/><\/span>the major axis is horizontal.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6360cc63b7ed131097795ca2761a48d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#61;&plusmn;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30025bdfbf02958404d8b4dda1b37b5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;&plusmn;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">The distance from the center to the vertices is 3.<span data-type=\"newline\"><br \/><\/span>The distance from the center to the endpoints of the<span data-type=\"newline\"><br \/><\/span>minor axis is 2.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the ellipse.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871925897\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872516051\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872456046\">\n<div data-type=\"problem\" id=\"fs-id1163872403700\">\n<p id=\"fs-id1163872541264\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6eaadd843e8ff93a2e6c6df8f7508b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"149\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872643398\"><span data-type=\"media\" id=\"fs-id1163871993543\" data-alt=\"This graph shows an ellipse with center at (negative 3, 5), vertices at (negative 3, 9) and (negative 3, 1) and endpoints of minor axis at (negative 5, 5) and (negative 1, 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center at (negative 3, 5), vertices at (negative 3, 9) and (negative 3, 1) and endpoints of minor axis at (negative 5, 5) and (negative 1, 5).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872769827\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163871859210\">\n<div data-type=\"problem\" id=\"fs-id1163871617684\">\n<p id=\"fs-id1163872545739\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-691e8392e8a0ecf2f3fc09ebaafec543_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"149\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872503302\"><span data-type=\"media\" id=\"fs-id1163872652811\" data-alt=\"This graph shows an ellipse with center at 1, negative 3, vertices at (negative 4, negative 3) and (6, negative 3) and endpoints of minor axis at 1, 1) and (negative 1, negative 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_307_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center at 1, negative 3, vertices at (negative 4, negative 3) and (6, negative 3) and endpoints of minor axis at 1, 1) and (negative 1, negative 7).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872033617\">If we look at the equations of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5980e076b60afda408a49959150c832d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c860310c86ce0d73755f2101d4775c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -6px;\" \/> we see that they are both ellipses with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9418f76b7fb8efbd61d4b14b3df06bad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a304623b8c0b2dd875899e936bcdbc4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> So they will have the same size and shape. They are different in that they do not have the same center.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872439917\" data-alt=\"The equation in the first figure is x squared upon 9 plus y squared upon 4 equals 1. Here, a is 3 and b is 2. The ellipse is graphed with center at (0, 0). The equation on the right is open parentheses x minus 3 close parentheses squared upon 9 plus open parentheses y minus 1 close parentheses squared upon 4 equals 1. Here, too, a is 3 and b is 2, but the center is (3, 1). The ellipse is shown on the same graph along with the first ellipse. The center is shown to have moved 3 units right and 1 unit up.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equation in the first figure is x squared upon 9 plus y squared upon 4 equals 1. Here, a is 3 and b is 2. The ellipse is graphed with center at (0, 0). The equation on the right is open parentheses x minus 3 close parentheses squared upon 9 plus open parentheses y minus 1 close parentheses squared upon 4 equals 1. Here, too, a is 3 and b is 2, but the center is (3, 1). The ellipse is shown on the same graph along with the first ellipse. The center is shown to have moved 3 units right and 1 unit up.\" \/><\/span><\/p>\n<p id=\"fs-id1163872546473\">\n<p id=\"fs-id1163871879747\">Notice in the graph above that we could have graphed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcc7e77e53ee242e07928690a4aecdb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/> by translations. We moved the original ellipse to the right 3 units and then up 1 unit.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872872391\" data-alt=\"This graph shows an ellipse translated from center (0, 0) to center (3, 1). The center has moved 3 units right and 1 unit up. The original ellipse has vertices at (negative 3, 0) and (3, 0) and endpoint of minor axis at (negative 2, 0) and (2, 0). The translated ellipse has vertices at (0, 1) and (6, 1) and endpoints of minor axis at (3, negative 1) and (3, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_013_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse translated from center (0, 0) to center (3, 1). The center has moved 3 units right and 1 unit up. The original ellipse has vertices at (negative 3, 0) and (3, 0) and endpoint of minor axis at (negative 2, 0) and (2, 0). The translated ellipse has vertices at (0, 1) and (6, 1) and endpoints of minor axis at (3, negative 1) and (3, 3).\" \/><\/span><\/p>\n<p id=\"fs-id1163872436095\">In the next example we will use the translation method to graph the ellipse.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872610918\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872562232\">\n<div data-type=\"problem\" id=\"fs-id1163872467428\">\n<p id=\"fs-id1163872566832\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90f2604643cf9c6967af3a1f0fdb9ab8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/> by translation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872514246\">\n<p id=\"fs-id1163872743794\">This ellipse will have the same size and shape as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09673539e874b0ff0abbd0d3fe9ded6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/> whose center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> We graph this ellipse first.<span data-type=\"newline\"><br \/><\/span> <\/p>\n<table id=\"fs-id1163872740608\" class=\"unnumbered unstyled can-break\" summary=\"The center is 0, 0). Since 16 is greater than 9, the major axis is horizontal. a squared is 16, so a is plus or minus 4. The vertices are (4, 0) and (negative 4, 0). b squared is 9 so b is plus or minus 3. The endpoints of the minor axis are (0, 3) and (0, negative 3). Graph the ellipse. The original equation is in standard form where h is minus 4 and k is 6. The center of the translated ellipse will be (negative 4, 6). We translate the graph of the first ellipse four units to the left and then up 6 units. Verify that the center is (negative 4, 6). The new ellipse has the desired equation.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f8855fa695d569a1cb586b456d62a24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#62;&#57;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -4px;\" \/> the major axis is horizontal.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d48b740446b4dfb00b2b8fffa30e8a2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#44;&#97;&#61;&plusmn;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fd97483c67023bf26c4a774fbbd4e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;&plusmn;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">The vertices are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e4cd56d89d31e40a781c004db11511a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>The endpoints of the minor axis are<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bfbf25347361f1e56619b1954e29057_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the ellipse.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871619081\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The original equation is in standard form,<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-155bd1abd30882967a1a7f5db788ecb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"151\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff7e3348562c61ed7bb4ed1c524120a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"166\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The ellipse is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">The center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bde61412db43aad12121cf37aa47c66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">We translate the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09673539e874b0ff0abbd0d3fe9ded6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/> four<span data-type=\"newline\"><br \/><\/span>units to the left and then up 6 units.<span data-type=\"newline\"><br \/><\/span>Verify that the center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bde61412db43aad12121cf37aa47c66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>The new ellipse is the ellipse whose equation<span data-type=\"newline\"><br \/><\/span>is<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d0456b788c5a7d3324e01be282c9457_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"149\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872534802\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163866197682\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872502349\">\n<div data-type=\"problem\" id=\"fs-id1163872697273\">\n<p id=\"fs-id1163872516184\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98a9be09ff3a4fca69ec0d49cbde03ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/> by translation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872388852\"><span data-type=\"media\" id=\"fs-id1163872628524\" data-alt=\"This graph shows an ellipse with center (5, negative 4), vertices (2, negative 4) and (8, negative 4) and endpoints of minor axis (5, negative 2) and (5, negative 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (5, negative 4), vertices (2, negative 4) and (8, negative 4) and endpoints of minor axis (5, negative 2) and (5, negative 6).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872537590\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872511084\">\n<div data-type=\"problem\" id=\"fs-id1163872569742\">\n<p id=\"fs-id1163872013766\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-117d9482aa5809b4b74c5a611daa57fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/> by translation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872037146\"><span data-type=\"media\" id=\"fs-id1163872426936\" data-alt=\"This graph shows an ellipse with center (negative 6, negative 2), vertices (negative 6, 3) and (negative 6, negative 7) and endpoints of minor axis (negative 10, negative 2), and (negative 2, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_309_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 6, negative 2), vertices (negative 6, 3) and (negative 6, negative 7) and endpoints of minor axis (negative 10, negative 2), and (negative 2, negative 2).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872666626\">When an equation has both an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> and a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/> with different coefficients, we verify that it is an ellipsis by putting it in standard form. We will then be able to graph the equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872570378\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163872721091\">\n<div data-type=\"problem\" id=\"fs-id1163872836545\">\n<p id=\"fs-id1163871979313\">Write the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1130f3086f4652a28745b5027874da50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#50;&#52;&#121;&#43;&#50;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"227\" style=\"vertical-align: -4px;\" \/> in standard form and graph.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872403290\">\n<p id=\"fs-id1163872837027\">We put the equation in standard form by completing the squares in both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<span data-type=\"newline\"><br \/><\/span> <\/p>\n<table class=\"unnumbered unstyled can-break\" summary=\"Rewrite the equation grouping the x terms and y terms. Make the coefficients of x squared and y squared equal to 1. We get open parentheses x squared minus 4 x plus close parentheses plus 4 open parentheses y squared plus 6y plus close parentheses equals minus 24. Complete the squares by adding 4 to the first term and 9 to the second term. The right side becomes minus 24 plus 4 plus 36. Write as binomial squares open parentheses x minus 2 close parentheses squared plus 4 open parentheses y plus 3 close parentheses squared equals 16. Divide both sides by 16 to get 1 on the right. The equation is in standard form with h equal to 2 and k equal to minus 3. The center is (2, negative 3). Since 16 is greater than 4 and is in the x squared term, the major axis is horizontal. a squared is 16, so a is plus or minus 4. The distance from the center to the vertices is 4. b squared is 4, so b is plus or minus 2. The distance from the center to the endpoints of the minor axis is 2. Graph the ellipse. It will have the points (2, negative 1), (2, negative 5), (6, negative 3) and (negative 2, negative 3).\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1130f3086f4652a28745b5027874da50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#50;&#52;&#121;&#43;&#50;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"227\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite grouping the <em data-effect=\"italics\">x<\/em> terms and <em data-effect=\"italics\">y<\/em> terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872694288\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Make the coefficients of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/> equal 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872544205\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Complete the squares.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872479153\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write as binomial squares.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872555298\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide both sides by 16 to get 1 on the right.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872623363\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163871781907\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The equation is in standard form,<span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdc35d2c510f0d6e68352ba88eea5b4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"146\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872499717\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The ellipse is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">The center is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb76d278ef8bf9c035a41ef898b53f04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee90a82eb157fe207819bba51733883b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#62;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: -1px;\" \/> and 16 is in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> term,<span data-type=\"newline\"><br \/><\/span>the major axis is horizontal.<span data-type=\"newline\"><br \/><\/span>\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d48b740446b4dfb00b2b8fffa30e8a2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#44;&#97;&#61;&plusmn;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2d17ff350db14340399b5fdfbbed6d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;&plusmn;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"bottom\" data-align=\"left\">The distance from the center to the vertices is 4.<span data-type=\"newline\"><br \/><\/span>The distance from the center to the endpoints of<span data-type=\"newline\"><br \/><\/span>the minor axis is 2.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the ellipse.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163872564752\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_015h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872611551\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872012760\">\n<div data-type=\"problem\" id=\"fs-id1163872511507\">\n<p id=\"fs-id1163872841104\"><span class=\"token\">\u24d0<\/span> Write the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6be61d4330b0eef92b45b3669e112d31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#51;&#50;&#121;&#43;&#51;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"245\" style=\"vertical-align: -4px;\" \/> in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872464984\">\n<p id=\"fs-id1163872456532\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9d2970ae63823ca2866e1f1d0455d81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872529490\" data-alt=\"This graph shows an ellipse with center (negative 1, 4), vertices minus (1, 1) and (negative 1, 7) and endpoints of minor axis approximately (negative 3.5, 4) and (approximately 1.5, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 1, 4), vertices minus (1, 1) and (negative 1, 7) and endpoints of minor axis approximately (negative 3.5, 4) and (approximately 1.5, 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872435902\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872850405\">\n<div data-type=\"problem\" id=\"fs-id1163872467826\">\n<p id=\"fs-id1163872660171\"><span class=\"token\">\u24d0<\/span> Write the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68076f4632711a85d9634ba0ea4a379e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#120;&#45;&#54;&#121;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -4px;\" \/> in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872409334\">\n<p id=\"fs-id1163872197218\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f552e667f763ccd55c9ce9e915f1e943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872015767\" data-alt=\"This graph shows an ellipse with center (2, 3), vertices (2, negative 1) and (2, 7) and endpoints of minor axis (0, 3) and (4, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_311_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (2, 3), vertices (2, negative 1) and (2, 7) and endpoints of minor axis (0, 3) and (4, 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1171792588082\">\n<h4 data-type=\"title\">Solve Application with Ellipses<\/h4>\n<p id=\"fs-id1163872447476\">The orbits of the planets around the sun follow elliptical paths.<\/p>\n<div data-type=\"example\" id=\"fs-id1163872840276\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163871987196\">\n<div data-type=\"problem\" id=\"fs-id1163872513963\">\n<p id=\"fs-id1163872529927\">Pluto (a dwarf planet) moves in an elliptical orbit around the Sun. The closest Pluto gets to the Sun is approximately 30 astronomical units (AU) and the furthest is approximately 50 AU. The Sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of Pluto.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872545123\" data-alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 40, 0) and (40, 0). The sun is shown at point (10, 0). This is 30 units from the right vertex and 50 units from the left vertex.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_016_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 40, 0) and (40, 0). The sun is shown at point (10, 0). This is 30 units from the right vertex and 50 units from the left vertex.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163868248157\">\n<p id=\"fs-id1163866196994\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-beedfa05acda3de01ad5293fb20c1b7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#114;&#101;&#99;&#111;&#103;&#110;&#105;&#122;&#101;&#32;&#116;&#104;&#105;&#115;&#32;&#97;&#115;&#32;&#97;&#110;&#32;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#32;&#116;&#104;&#97;&#116;&#32;&#105;&#115;&#32;&#99;&#101;&#110;&#116;&#101;&#114;&#101;&#100;&#32;&#97;&#116;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#105;&#103;&#105;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#109;&#97;&#106;&#111;&#114;&#32;&#97;&#120;&#105;&#115;&#32;&#105;&#115;&#32;&#104;&#111;&#114;&#105;&#122;&#111;&#110;&#116;&#97;&#108;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#32;&#102;&#114;&#111;&#109;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#99;&#101;&#110;&#116;&#101;&#114;&#32;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#118;&#101;&#114;&#116;&#101;&#120;&#32;&#105;&#115;&#32;&#52;&#48;&#44;&#32;&#119;&#101;&#32;&#107;&#110;&#111;&#119;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#61;&#52;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#115;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#48;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#48;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#109;&#105;&#110;&#111;&#114;&#32;&#97;&#120;&#105;&#115;&#32;&#105;&#115;&#32;&#118;&#101;&#114;&#116;&#105;&#99;&#97;&#108;&#32;&#98;&#117;&#116;&#32;&#116;&#104;&#101;&#32;&#101;&#110;&#100;&#32;&#112;&#111;&#105;&#110;&#116;&#115;&#32;&#97;&#114;&#101;&#110;&#39;&#116;&#32;&#103;&#105;&#118;&#101;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#101;&#32;&#119;&#105;&#108;&#108;&#32;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#108;&#111;&#99;&#97;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#83;&#117;&#110;&#46;&#32;&#83;&#105;&#110;&#99;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#110;&#32;&#105;&#115;&#32;&#97;&#32;&#102;&#111;&#99;&#117;&#115;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#32;&#97;&#116;&#32;&#116;&#104;&#101;&#32;&#112;&#111;&#105;&#110;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#101;&#32;&#107;&#110;&#111;&#119;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#61;&#49;&#48;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#105;&#115;&#32;&#116;&#111;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#52;&#48;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#48;&#48;&#45;&#49;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#53;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#48;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#53;&#48;&#48;&#125;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"222\" width=\"757\" style=\"vertical-align: -108px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872652849\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872836806\">\n<div data-type=\"problem\" id=\"fs-id1163872023288\">\n<p id=\"fs-id1163871756678\">A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately 20 AU and the furthest is approximately 30 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163871975450\" data-alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 25, 0) and (25, 0). The sun is shown at point (5, 0). This is 20 units from the right vertex and 30 units from the left vertex.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_017_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 25, 0) and (25, 0). The sun is shown at point (5, 0). This is 20 units from the right vertex and 30 units from the left vertex.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872516912\">\n<p id=\"fs-id1163872092443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7997accd1d999d918538bc9a7a679112_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#54;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#54;&#48;&#48;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"100\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872436182\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163872740817\">\n<div data-type=\"problem\" id=\"fs-id1163872432940\">\n<p id=\"fs-id1163872464205\">A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately 20 AU and the furthest is approximately 50 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872514058\" data-alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 35, 0) and (35, 0). The sun is shown at point (15, 0). This is 20 units from the right vertex and 50 units from the left vertex.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_018_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0) and vertices (negative 35, 0) and (35, 0). The sun is shown at point (15, 0). This is 20 units from the right vertex and 50 units from the left vertex.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872665959\">\n<p id=\"fs-id1163872713392\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-890686eecf92684f149de69121806461_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#50;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#48;&#48;&#48;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"114\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163872096564\" class=\"media-2\">\n<p id=\"fs-id1163872718869\">Access these online resources for additional instructions and practice with ellipses.<\/p>\n<ul id=\"fs-id1163872503067\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37graphellipse1\">Conic Sections: Graphing Ellipses Part 1<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37graphellipse2\">Conic Sections: Graphing Ellipses Part 2<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37eqellipse\">Equation for Ellipse From Graph<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163872697768\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1163872096217\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Ellipse:<\/strong> An <strong data-effect=\"bold\">ellipse<\/strong> is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the ellipse.<span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1163872574755\" data-alt=\"This figure shows two ellipses. In each, two points within the ellipse are labeled foci. A line drawn through the foci intersects the ellipse in two points. Each point is labeled a vertex. In The figure on the left, the segment connecting the vertices is called the major axis. A segment perpendicular to the major axis that passes through its midpoint and intersects the ellipse in two points is labeled minor axis. The major axis is longer than the minor axis. In The figure on the right, the segment through the foci, connecting the vertices is shorter and is labeled minor axis. Its midpoint is labeled center.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_020_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows two ellipses. In each, two points within the ellipse are labeled foci. A line drawn through the foci intersects the ellipse in two points. Each point is labeled a vertex. In The figure on the left, the segment connecting the vertices is called the major axis. A segment perpendicular to the major axis that passes through its midpoint and intersects the ellipse in two points is labeled minor axis. The major axis is longer than the minor axis. In The figure on the right, the segment through the foci, connecting the vertices is shorter and is labeled minor axis. Its midpoint is labeled center.\" \/><\/span><span data-type=\"newline\"><br \/><\/span> If we draw a line through the foci intersects the ellipse in two points\u2014each is called a <strong data-effect=\"bold\">vertex<\/strong> of the ellipse.<span data-type=\"newline\"><br \/><\/span> The segment connecting the vertices is called the <strong data-effect=\"bold\">major axis<\/strong>.<span data-type=\"newline\"><br \/><\/span> The midpoint of the segment is called the <strong data-effect=\"bold\">center<\/strong> of the ellipse.<span data-type=\"newline\"><br \/><\/span> A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the <strong data-effect=\"bold\">minor axis<\/strong>.<\/li>\n<li><strong data-effect=\"bold\">Standard Form of the Equation an Ellipse with Center<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-231f1e89a99c6a1f34acbd871706dfe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> The standard form of the equation of an ellipse with center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-825405fc63416ad0c306970366e996d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> is<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1163872624303\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4100e34708a781839b2e089b9992ca73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/div>\n<p><span data-type=\"newline\"><br \/><\/span> The <em data-effect=\"italics\">x<\/em>-intercepts are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-914e9523c72b2e0b3d6961c80651140f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> The <em data-effect=\"italics\">y<\/em>-intercepts are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05dbabd42fe84ba97e673be0628c5974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42fd44aa7d2ea598ea3ef7b69187f6b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><strong data-effect=\"bold\">How to an Ellipse with Center<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>\n<ol id=\"fs-id1163871923665\" type=\"1\" class=\"stepwise\">\n<li>Write the equation in standard form.<\/li>\n<li>Determine whether the major axis is horizontal or vertical.<\/li>\n<li>Find the endpoints of the major axis.<\/li>\n<li>Find the endpoints of the minor axis<\/li>\n<li>Sketch the ellipse.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Standard Form of the Equation an Ellipse with Center<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64d58217a4fc7663d01902d08028efa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -4px;\" \/> The standard form of the equation of an ellipse with center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02b8dd65a353ed3cc1e39e08436cf52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> is<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1163872575490\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdc35d2c510f0d6e68352ba88eea5b4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"146\" style=\"vertical-align: -7px;\" \/><\/div>\n<p><span data-type=\"newline\"><br \/><\/span> When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b069cd4e45edfcf03bedc4812930311_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#98;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/> the major axis is horizontal so the distance from the center to the vertex is <em data-effect=\"italics\">a<\/em>.<span data-type=\"newline\"><br \/><\/span> When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-241db334cff6d4114e26d1794f470af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#62;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> the major axis is vertical so the distance from the center to the vertex is <em data-effect=\"italics\">b<\/em>.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163872769241\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1163871876800\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1163872019091\"><strong data-effect=\"bold\">Graph an Ellipse with Center at the Origin<\/strong><\/p>\n<p id=\"fs-id1163872014059\">In the following exercises, graph each ellipse.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872829148\">\n<div data-type=\"problem\" id=\"fs-id1163871883835\">\n<p id=\"fs-id1163871890082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24c6a0e61582bcbdd04b9fed7f35cd69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872557100\"><span data-type=\"media\" id=\"fs-id1163872534088\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (2, 0) and (negative 2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (2, 0) and (negative 2, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871997735\">\n<div data-type=\"problem\" id=\"fs-id1163872337696\">\n<p id=\"fs-id1163872569434\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caf7f5d1e1008ae2c5409d331d8a3043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872709615\">\n<div data-type=\"problem\" id=\"fs-id1163872092335\">\n<p id=\"fs-id1163872427624\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-533cbe310cb31865c8d263acb2254d3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872742674\"><span data-type=\"media\" id=\"fs-id1163865785818\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (5, 0) and (negative 5, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (5, 0) and (negative 5, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872110892\">\n<div data-type=\"problem\" id=\"fs-id1163872427472\">\n<p id=\"fs-id1163872103649\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33039d8a5741a553fc8b1c22e84031b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872514193\">\n<div data-type=\"problem\" id=\"fs-id1163871980336\">\n<p id=\"fs-id1163872768352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d758bf9ad4bd5e19814f8d00990ee87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871926860\"><span data-type=\"media\" id=\"fs-id1163868454052\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_316_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872531832\">\n<div data-type=\"problem\" id=\"fs-id1163872804309\">\n<p id=\"fs-id1163872804311\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-981d69a739d9d3a7d6c28681bebc5467_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872611075\">\n<div data-type=\"problem\" id=\"fs-id1163872560474\">\n<p id=\"fs-id1163872560476\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ab7a36f0075ec1e0e909a70e3549b67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872531758\"><span data-type=\"media\" id=\"fs-id1163865733298\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 2) and (0, negative 2) and endpoints of minor axis (1, 0) and (negative 1, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_318_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 2) and (0, negative 2) and endpoints of minor axis (1, 0) and (negative 1, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872768883\">\n<div data-type=\"problem\" id=\"fs-id1163872822384\">\n<p id=\"fs-id1163872542678\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7877b41a83eab5c015e4a307a7b00e55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"87\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163865848463\">\n<div data-type=\"problem\" id=\"fs-id1163865848465\">\n<p id=\"fs-id1163872567928\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6986ae4e74070667f81f5c59ab374612_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872511469\"><span data-type=\"media\" id=\"fs-id1163872569570\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_320_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872536189\">\n<div data-type=\"problem\" id=\"fs-id1163872024666\">\n<p id=\"fs-id1163872024668\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d2704801c5f75b76e0a63301dcdc330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871868040\">\n<div data-type=\"problem\" id=\"fs-id1163872719600\">\n<p id=\"fs-id1163872705504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2b8725d10a46632dccb6da6b97c8bd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#55;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871935344\"><span data-type=\"media\" id=\"fs-id1163871930792\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_322_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872768062\">\n<div data-type=\"problem\" id=\"fs-id1163872768064\">\n<p id=\"fs-id1163872024700\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28835fbf027b5c8f2a7d79d95d3af19a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872554720\"><strong data-effect=\"bold\">Find the Equation of an Ellipse with Center at the Origin<\/strong><\/p>\n<p id=\"fs-id1163872394595\">In the following exercises, find the equation of the ellipse shown in the graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163868451657\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163868451660\">\n<p id=\"fs-id1163872420236\"><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872420237\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (negative 3, 0) and (3, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (negative 3, 0) and (3, 0).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872663250\">\n<p id=\"fs-id1163872600785\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caf7f5d1e1008ae2c5409d331d8a3043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871710691\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163871866782\">\n<p id=\"fs-id1163871531511\"><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163871531512\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_202_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871874103\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872521543\">\n<p id=\"fs-id1163872521545\"><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872110551\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 4) and (0, negative 4) and endpoints of minor axis (negative 3, 0) and (3, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_203_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 4) and (0, negative 4) and endpoints of minor axis (negative 3, 0) and (3, 0).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872514670\">\n<p id=\"fs-id1163872709202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e36a043d7d76e04d8da0d32fd9cd6cde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872768251\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163871992932\">\n<p id=\"fs-id1163871934957\"><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163871934958\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 4, 0) and (4, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_204_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 4, 0) and (4, 0).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163872015114\"><strong data-effect=\"bold\">Graph an Ellipse with Center Not at the Origin<\/strong><\/p>\n<p>In the following exercises, graph each ellipse.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872418395\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872418397\">\n<p id=\"fs-id1163871987398\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07daa4f92f1a20143b1feb763d7f97c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871994520\"><span data-type=\"media\" id=\"fs-id1163872096387\" data-alt=\"This graph shows an ellipse with center (negative 1, negative 6, vertices (negative 1, negative 1) and (negative 1, negative 11) and endpoints of minor axis (negative 3, negative 6) and (1, negative 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_324_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 1, negative 6, vertices (negative 1, negative 1) and (negative 1, negative 11) and endpoints of minor axis (negative 3, negative 6) and (1, negative 6).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872570186\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872533709\">\n<p id=\"fs-id1163872533711\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e97dfdb9122c6c4999b485acdba754a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872842208\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872612895\">\n<p id=\"fs-id1163872612897\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae2fd8fe3eb731d7dcc3a2982ae4b111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872302554\"><span data-type=\"media\" id=\"fs-id1163872108528\" data-alt=\"This graph shows an ellipse with center (negative 4, 2, vertices (negative 4, 5) and (negative 4, negative 1) and endpoints of minor axis (3, 1) and (negative 6, 2) and (negative 2, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_326_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 4, 2, vertices (negative 4, 5) and (negative 4, negative 1) and endpoints of minor axis (3, 1) and (negative 6, 2) and (negative 2, 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872448776\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872720576\">\n<p id=\"fs-id1163872720578\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68e48288de5229ced0bfd50a62fc8066_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163871932647\">In the following exercises, graph each equation by translation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163871783180\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872022197\">\n<p id=\"fs-id1163872022200\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ad607f8f8ce35fbbc6429226912622d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872708891\"><span data-type=\"media\" id=\"fs-id1163872799939\" data-alt=\"This graph shows an ellipse with center (3, 7), vertices (3, 2) and (3, 12), and endpoints of minor axis (1, 7) and (5, 7).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_328_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (3, 7), vertices (3, 2) and (3, 12), and endpoints of minor axis (1, 7) and (5, 7).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871882005\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872403279\">\n<p id=\"fs-id1163872403281\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0f6d0ec81e68fa6db62553912587a03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871927127\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872453024\">\n<p id=\"fs-id1163871985605\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cd7b3c484c932a73c7252a7df8a4dd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872630902\"><span data-type=\"media\" id=\"fs-id1163872444389\" data-alt=\"This graph shows an ellipse with center (5, negative 4), vertices (5, 1) and (5, negative 9) and endpoints of minor axis (2, negative 4) and (8, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_330_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (5, negative 4), vertices (5, 1) and (5, negative 9) and endpoints of minor axis (2, negative 4) and (8, negative 4).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872510998\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872014761\">\n<p id=\"fs-id1163871378181\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-332b1b134ce0fcf71d27240f5d6366cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872833573\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872693587\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872573326\">\n<p id=\"fs-id1163872553270\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bd234405ebc8406de7772313df36aba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#48;&#120;&#45;&#53;&#52;&#121;&#45;&#52;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"262\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872660997\">\n<p id=\"fs-id1163872034704\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8618b36f3ef837261174968276d4f244_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163871934818\" data-alt=\"This graph shows an ellipse with center (2, 3), vertices (2, negative 2) and (2, 8) and endpoints of minor axis (negative 1, 3) and (5, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_332_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (2, 3), vertices (2, negative 2) and (2, 8) and endpoints of minor axis (negative 1, 3) and (5, 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872841555\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163871933662\">\n<p id=\"fs-id1163872743279\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-090fee8244285fa2a0d041049b6edd0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#48;&#48;&#121;&#43;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"245\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871704288\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872397377\">\n<p id=\"fs-id1163872639909\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1131255a83b88943ee69c01405e81cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#120;&#45;&#54;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"205\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163868452715\">\n<p id=\"fs-id1163872446383\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47c5d34fed18b1a1f96c04a43b32509c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"116\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872093634\" data-alt=\"This graph shows an ellipse with center (3, 0), vertices (negative 2, 0) and (8, 0) and endpoints of minor axis (3, 2) and (3, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_334_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (3, 0), vertices (negative 2, 0) and (8, 0) and endpoints of minor axis (3, 2) and (3, negative 2).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872515061\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872515063\">\n<p id=\"fs-id1163872095366\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed9d7f2b85b31a8511421f432d69767d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#54;&#121;&#43;&#49;&#54;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872469152\">In the following exercises, graph the equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163872400116\">\n<div data-type=\"problem\" id=\"fs-id1163872400118\">\n<p id=\"fs-id1163872693934\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f12e068b6e2d9d80f4f7ea02b83c67e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872093400\"><span data-type=\"media\" id=\"fs-id1163871349264\" data-alt=\"This graph shows a parabola with vertex (2, 1) and y intercepts (0, 0) and (2, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_336_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a parabola with vertex (2, 1) and y intercepts (0, 0) and (2, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163866047685\">\n<div data-type=\"problem\" id=\"fs-id1163872467848\">\n<p id=\"fs-id1163872559529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4846d3ba91620499c56c560ff0529ae1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872600142\">\n<div data-type=\"problem\" id=\"fs-id1163872600144\">\n<p id=\"fs-id1163872833517\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b8c8ade0ae5731f33180708258a3d64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872243575\"><span data-type=\"media\" id=\"fs-id1163872742897\" data-alt=\"This graph shows a circle with center (negative 5, negative 2) and a radius of 2 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_338_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows a circle with center (negative 5, negative 2) and a radius of 2 units.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872664239\">\n<div data-type=\"problem\" id=\"fs-id1163872646173\">\n<p id=\"fs-id1163872666370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efc32fd0dcb0d7732f6656b8509d192a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871756540\">\n<div data-type=\"problem\" id=\"fs-id1163872034683\">\n<p id=\"fs-id1163872576509\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3f0af23e8c24a9bce4eb546d8be3256_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871567468\"><span data-type=\"media\" id=\"fs-id1163865730465\" data-alt=\"This graph shows an ellipse with center (negative 3, negative 1), vertices (1, negative 1) and (negative 7, negative 1) and endpoints of minor axis (negative 3, 1) and (negative 3, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_340_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (negative 3, negative 1), vertices (1, negative 1) and (negative 7, negative 1) and endpoints of minor axis (negative 3, 1) and (negative 3, negative 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872557744\">\n<div data-type=\"problem\" id=\"fs-id1163872630243\">\n<p id=\"fs-id1163866226794\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3d2a9911b42757757d09aea8d8fe6b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872629661\">\n<div data-type=\"problem\" id=\"fs-id1163872629663\">\n<p id=\"fs-id1163871930849\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-533cbe310cb31865c8d263acb2254d3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872642609\"><span data-type=\"media\" id=\"fs-id1163866198996\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 5, 0) and (5, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_342_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 5, 0) and (5, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871866286\">\n<div data-type=\"problem\" id=\"fs-id1163871866288\">\n<p id=\"fs-id1163872462579\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d752759191a32a666cf0385e286a62f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871908392\">\n<div data-type=\"problem\" id=\"fs-id1163871908394\">\n<p id=\"fs-id1163871859292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-358bac6ea7b08a21b71ed22199d52e58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163868434169\"><span data-type=\"media\" id=\"fs-id1163872532047\" data-alt=\"This graph shows circle with center (0, 0) and with radius 8 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_344_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows circle with center (0, 0) and with radius 8 units.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871870368\">\n<div data-type=\"problem\" id=\"fs-id1163871870370\">\n<p id=\"fs-id1163872799994\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caf7f5d1e1008ae2c5409d331d8a3043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872560516\">\n<div data-type=\"problem\" id=\"fs-id1163872840603\">\n<p id=\"fs-id1163872840605\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad1ac3f62be96cfdc25626aeb4c3625d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871731254\"><span data-type=\"media\" id=\"fs-id1163872033812\" data-alt=\"This graph shows upward opening parabola. Its vertex has an x value of slightly less than 0 and a y value of slightly less than minus 1. A point on it is approximately at (negative 1, 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_346_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows upward opening parabola. Its vertex has an x value of slightly less than 0 and a y value of slightly less than minus 1. A point on it is approximately at (negative 1, 3).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872940076\">\n<div data-type=\"problem\" id=\"fs-id1163872729685\">\n<p id=\"fs-id1163872729687\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-214565c8e809e35640044ec29e80d98f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163872472137\"><strong data-effect=\"bold\">Solve Application with Ellipses<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1163871987462\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163871987464\">\n<p id=\"fs-id1163872562521\">A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately 10 AU and the furthest is approximately 30 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872562525\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 20, 0) and (20, 0). The sun is shown at point (10, 0), which is 30 units from the left vertex and 10 units from the right vertex.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_205_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 20, 0) and (20, 0). The sun is shown at point (10, 0), which is 30 units from the left vertex and 10 units from the right vertex.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872548932\">\n<p id=\"fs-id1163872105286\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9679b127c1a12e1b70fcd6f929d0858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#48;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#48;&#48;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"100\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872435929\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163872435931\">\n<p id=\"fs-id1163872404828\">A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately 10 AU and the furthest is approximately 70 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872404831\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 40, 0) and (40, 0). The sun is shown at point (30, 0), which is 70 units from the left vertex and 10 units from the right vertex.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_206_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 40, 0) and (40, 0). The sun is shown at point (30, 0), which is 70 units from the left vertex and 10 units from the right vertex.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163871895792\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163871895794\">\n<p id=\"fs-id1163872556219\">A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 15 AU and the furthest is approximately 85 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872556221\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 50, 0) and (50, 0). The sun is shown at point (35, 0), which is 85 units from the left vertex and 15 units from the right vertex.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_207_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 50, 0) and (50, 0). The sun is shown at point (35, 0), which is 85 units from the left vertex and 15 units from the right vertex.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1163872561506\">\n<p id=\"fs-id1163872561508\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb74b19d4417fc4a4743ef12153cd4f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#48;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#50;&#55;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"114\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872748963\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163871704679\">\n<p id=\"fs-id1163871704681\">A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately 15 AU and the furthest is approximately 95 AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872599043\" data-alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 55, 0) and (55, 0). The sun is shown at point (40, 0), which is 95 units from the left vertex and 15 units from the right vertex.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_208_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This graph shows an ellipse with center (0, 0), vertices (negative 55, 0) and (55, 0). The sun is shown at point (40, 0), which is 95 units from the left vertex and 15 units from the right vertex.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1163863474708\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1163871993197\">\n<div data-type=\"problem\" id=\"fs-id1163871993199\">\n<p id=\"fs-id1163871977038\">In your own words, define an ellipse and write the equation of an ellipse centered at the origin in standard form. Draw a sketch of the ellipse labeling the center, vertices and major and minor axes.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163871977040\">\n<p id=\"fs-id1163872800713\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872564055\">\n<div data-type=\"problem\" id=\"fs-id1163872011994\">\n<p id=\"fs-id1163872011996\">Explain in your own words how to get the axes from the equation in standard form.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872713558\">\n<div data-type=\"problem\" id=\"fs-id1163872713560\">\n<p id=\"fs-id1163872546052\">Compare and contrast the graphs of the equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2d250acae6b25bd89b6409d2e8516c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d2d652695b72a5e66922194dba71f4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163872713406\">\n<p id=\"fs-id1163872713408\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163872020384\">\n<div data-type=\"problem\" id=\"fs-id1163872435908\">\n<p id=\"fs-id1163872435910\">Explain in your own words, the difference between a vertex and a focus of the ellipse.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163872095843\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1163872402231\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163872548678\" data-alt=\"This table has 4 columns 4 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don&#x2019;t get it. The first columns has the following statements: graph an ellipse with center at the origin, find the equation of an ellipse with center at the origin, graph an ellipse with center not at the origin, solve applications with ellipses. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_03_209_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns 4 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don&#x2019;t get it. The first columns has the following statements: graph an ellipse with center at the origin, find the equation of an ellipse with center at the origin, graph an ellipse with center not at the origin, solve applications with ellipses. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1163871568158\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1163871993300\">\n<dt>ellipse<\/dt>\n<dd id=\"fs-id1163872747198\">An ellipse is all points in a plane where the sum of the distances from two fixed points is constant.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15494","chapter","type-chapter","status-publish","hentry"],"part":15253,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15494","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15494\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/15253"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15494\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=15494"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=15494"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=15494"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=15494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}