{"id":1591,"date":"2025-07-25T04:00:19","date_gmt":"2025-07-25T04:00:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1591"},"modified":"2026-03-25T04:24:35","modified_gmt":"2026-03-25T04:24:35","slug":"ellipses-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/ellipses-fresh-take\/","title":{"raw":"Ellipses: Fresh Take","rendered":"Ellipses: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write equations of ellipses in standard form.<\/li>\r\n \t<li>Graph ellipses.<\/li>\r\n \t<li>Solve applied problems involving ellipses.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Ellipses<\/h2>\r\nAn <strong>ellipse<\/strong> is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a <strong>focus<\/strong> (plural: <strong>foci<\/strong>) of the ellipse.We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03202213\/CNX_Precalc_Figure_10_01_0032.jpg\" alt=\"\" width=\"487\" height=\"560\" \/> Visual representation of how to draw an ellipse[\/caption]\r\n\r\nEvery ellipse has two axes of symmetry. The longer axis is called the <strong>major axis<\/strong>, and the shorter axis is called the <strong>minor axis<\/strong>. Each endpoint of the major axis is the <strong>vertex<\/strong> of the ellipse (plural: <strong>vertices<\/strong>), and each endpoint of the minor axis is a <strong>co-vertex<\/strong> of the ellipse. The <strong>center of an ellipse<\/strong> is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci.\r\n\r\nTo derive the equation of an\u00a0<span class=\"no-emphasis\">ellipse<\/span>\u00a0centered at the origin, we begin with the foci [latex](-c,0)[\/latex] and\u00a0[latex](-c,0)[\/latex]. The ellipse is the set of all points\u00a0[latex](x,y)[\/latex] such that the sum of the distances from\u00a0[latex](x,y)[\/latex] to the foci is constant, as shown in the figure below.\r\n\r\n[caption id=\"attachment_2327\" align=\"aligncenter\" width=\"530\"]<img class=\"wp-image-2327\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03202217\/CNX_Precalc_Figure_10_01_0142-300x169.jpg\" alt=\"\" width=\"530\" height=\"299\" \/> Ellipse with key features labeled[\/caption]\r\n<p id=\"fs-id1226529\">If [latex](a,0)[\/latex] is a\u00a0<span class=\"no-emphasis\">vertex<\/span>\u00a0of the ellipse, the distance from\u00a0[latex](-c,0)[\/latex] to [latex](a,0)[\/latex] is [latex]a-(-c)=a+c[\/latex]. The distance from [latex](c,0)[\/latex] to [latex](a,0)[\/latex] is [latex]a-c[\/latex]. The sum of the distances from the\u00a0<span class=\"no-emphasis\">foci<\/span>\u00a0to the vertex is<\/p>\r\n<p style=\"text-align: center;\">[latex](a+c)+(a-c)=2a[\/latex]<\/p>\r\n<p style=\"text-align: left;\">If [latex](x,y)[\/latex] is a point on the ellipse, then we can define the following variables:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}d_1&amp;=\\text{the distance from } (-c,0) \\text{ to } (x,y) \\\\ d_2&amp;= \\text{the distance from } (c,0) \\text{ to } (x,y) \\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">By the definition of an ellipse, [latex]d_1+d_2[\/latex] is constant for any point [latex](x,y)[\/latex] on the ellipse. We know that the sum of these distances is [latex]2a[\/latex] for the vertex [latex](a,0)[\/latex]. It follows that [latex]d_1+d_2=2a[\/latex] for any point on the ellipse. The derivation of the standard form of the equation of an ellipse relies on this relationship and the distance formula. The derivation is beyond the scope of this course, but the equation is:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1[\/latex]<\/p>\r\nfor an ellipse centered at the origin with its major axis on the\u00a0[latex]x[\/latex]-axis and\r\n<p style=\"text-align: center;\">[latex]\\dfrac{x^2}{b^2}+\\dfrac{y^2}{a^2}=1[\/latex]<\/p>\r\nfor an ellipse centered at the origin with its major axis on the [latex]y[\/latex]-axis.\r\n<h2>Writing Equations and Graphing Ellipses Centered at the Origin in Standard Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Standard Forms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal major axis: [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1[\/latex], where [latex]a &gt; b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical major axis: [latex]\\frac{x^2}{b^2} + \\frac{y^2}{a^2} = 1[\/latex], where [latex]a &gt; b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Important Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center: Always [latex](0, 0)[\/latex] for these forms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertices: [latex](\u00b1a, 0)[\/latex] for horizontal, [latex](0, \u00b1a)[\/latex] for vertical<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Co-vertices: [latex](0, \u00b1b)[\/latex] for horizontal, [latex](\u00b1b, 0)[\/latex] for vertical<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Foci: [latex](\u00b1c, 0)[\/latex] for horizontal, [latex](0, \u00b1c)[\/latex] for vertical, where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Axes Lengths:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Major axis: [latex]2a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Minor axis: [latex]2b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>From Equation to Graph<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify orientation (horizontal\/vertical) based on larger denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]a[\/latex] and [latex]b[\/latex] from equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] using [latex]c^2 = a^2 - b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot center, vertices, co-vertices, and foci<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sketch the ellipse through these points<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>From Points to Equation<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determine orientation from given points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]a[\/latex] and [latex]c[\/latex] from vertices and foci<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex] using [latex]b^2 = a^2 - c^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute values into appropriate standard form<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the ellipse that has vertices [latex]\\left(0,\\pm 8\\right)[\/latex] and foci\u00a0[latex](0,\\pm \\sqrt{5})[\/latex]?[reveal-answer q=\"997539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"997539\"][latex]x^2+\\dfrac{y^2}{16}=1[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph the ellipse given by the equation [latex]\\dfrac{{x}^{2}}{36}+\\dfrac{{y}^{2}}{4}=1[\/latex]. Identify and label the center, vertices, co-vertices, and foci.[reveal-answer q=\"796288\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"796288\"]center: [latex]\\left(0,0\\right)[\/latex]; vertices: [latex]\\left(\\pm 6,0\\right)[\/latex]; co-vertices: [latex]\\left(0,\\pm 2\\right)[\/latex]; foci: [latex]\\left(\\pm 4\\sqrt{2},0\\right)[\/latex]\r\n\r\n[caption id=\"attachment_3258\" align=\"aligncenter\" width=\"731\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02000914\/CNX_Precalc_Figure_10_01_0082.jpg\"><img class=\"wp-image-3258 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02000914\/CNX_Precalc_Figure_10_01_0082.jpg\" alt=\"\" width=\"731\" height=\"366\" \/><\/a> Ellipse with the center and key points labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph the ellipse given by the equation [latex]49{x}^{2}+16{y}^{2}=784[\/latex]. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.[reveal-answer q=\"866773\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"866773\"]Standard form: [latex]\\dfrac{{x}^{2}}{16}+\\dfrac{{y}^{2}}{49}=1[\/latex]; center: [latex]\\left(0,0\\right)[\/latex]; vertices: [latex]\\left(0,\\pm 7\\right)[\/latex]; co-vertices: [latex]\\left(\\pm 4,0\\right)[\/latex]; foci: [latex]\\left(0,\\pm \\sqrt{33}\\right)[\/latex]\r\n\r\n[caption id=\"attachment_3259\" align=\"aligncenter\" width=\"731\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02001223\/CNX_Precalc_Figure_10_01_0102.jpg\"><img class=\"wp-image-3259 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02001223\/CNX_Precalc_Figure_10_01_0102.jpg\" alt=\"\" width=\"731\" height=\"741\" \/><\/a> Ellipse with the center and key points labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbcdgabg-oZB69DY0q9A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/oZB69DY0q9A?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dbcdgabg-oZB69DY0q9A\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851181&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dbcdgabg-oZB69DY0q9A&amp;vembed=0&amp;video_id=oZB69DY0q9A&amp;video_target=tpm-plugin-dbcdgabg-oZB69DY0q9A\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Conic+Sections+-++The+Ellipse+part+2+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Ellipse part 2 of 2\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-chehfegb-azI5kALyiXs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/azI5kALyiXs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-chehfegb-azI5kALyiXs\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851279&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-chehfegb-azI5kALyiXs&amp;vembed=0&amp;video_id=azI5kALyiXs&amp;video_target=tpm-plugin-chehfegb-azI5kALyiXs\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Graph+an+Ellipse+with+Center+at+the+Origin+and+Horizontal+Major+Axis_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Graph an Ellipse with Center at the Origin and Horizontal Major Axis\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Writing Equations and Graphing Ellipses Not Centered at the Origin<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Standard Forms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal major axis: [latex]\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1[\/latex], where [latex]a &gt; b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical major axis: [latex]\\frac{(x-h)^2}{b^2} + \\frac{(y-k)^2}{a^2} = 1[\/latex], where [latex]a &gt; b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex](h, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertices: [latex](h \u00b1 a, k)[\/latex] for horizontal, [latex](h, k \u00b1 a)[\/latex] for vertical<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Co-vertices: [latex](h, k \u00b1 b)[\/latex] for horizontal, [latex](h \u00b1 b, k)[\/latex] for vertical<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Foci: [latex](h \u00b1 c, k)[\/latex] for horizontal, [latex](h, k \u00b1 c)[\/latex] for vertical, where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Axes Lengths:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Major axis: [latex]2a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Minor axis: [latex]2b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>From Equation to Graph<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify orientation (horizontal\/vertical) based on larger denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine center (h, k) from equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]a[\/latex] and [latex]b[\/latex] from equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] using [latex]c^2 = a^2 - b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot center, vertices, co-vertices, and foci<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sketch the ellipse through these points<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>From Points to Equation<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Determine orientation from given points<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find center using midpoint formula<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]a[\/latex] (half the distance between vertices)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] (distance from center to focus)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex] using [latex]b^2 = a^2 - c^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute values into appropriate standard form<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the ellipse that has vertices [latex]\\left(-3,3\\right)[\/latex] and [latex]\\left(5,3\\right)[\/latex] and foci [latex]\\left(1 - 2\\sqrt{3},3\\right)[\/latex] and [latex]\\left(1+2\\sqrt{3},3\\right)?[\/latex][reveal-answer q=\"256644\"]Show Solution[\/reveal-answer][hidden-answer a=\"256644\"][latex]\\dfrac{{\\left(x - 1\\right)}^{2}}{16}+\\dfrac{{\\left(y - 3\\right)}^{2}}{4}=1[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Graph the ellipse given by the equation [latex]\\dfrac{{\\left(x - 4\\right)}^{2}}{36}+\\dfrac{{\\left(y - 2\\right)}^{2}}{20}=1[\/latex]. Identify and label the center, vertices, co-vertices, and foci.[reveal-answer q=\"590423\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"590423\"]Center: [latex]\\left(4,2\\right)[\/latex]; vertices: [latex]\\left(-2,2\\right)[\/latex] and [latex]\\left(10,2\\right)[\/latex]; co-vertices: [latex]\\left(4,2 - 2\\sqrt{5}\\right)[\/latex] and [latex]\\left(4,2+2\\sqrt{5}\\right)[\/latex]; foci: [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(8,2\\right)[\/latex]\r\n\r\n[caption id=\"attachment_3260\" align=\"aligncenter\" width=\"731\"]<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02001521\/CNX_Precalc_Figure_10_01_0122.jpg\"><img class=\"wp-image-3260 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02001521\/CNX_Precalc_Figure_10_01_0122.jpg\" alt=\"\" width=\"731\" height=\"666\" \/><\/a> Ellipse with the center and key points labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gchbachg-tJmp1PJD9o8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tJmp1PJD9o8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gchbachg-tJmp1PJD9o8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851280&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gchbachg-tJmp1PJD9o8&amp;vembed=0&amp;video_id=tJmp1PJD9o8&amp;video_target=tpm-plugin-gchbachg-tJmp1PJD9o8\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Graph+an+Ellipse+with+Center+NOT+at+the+Origin+and+Horizontal+Major+Axis_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Graph an Ellipse with Center NOT at the Origin and Horizontal Major Axis\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Graphing an Ellipse in General Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">General Form: [latex]ax^2 + by^2 + cx + dy + e = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Goal: Transform to Standard Form [latex]\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Transformation Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Group like terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complete the square for [latex]x[\/latex] and [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Divide by constant term<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Features:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center [latex](h, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertices: [latex](h \u00b1 a, k)[\/latex] or [latex](h, k \u00b1 a)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Co-vertices: [latex](h, k \u00b1 b)[\/latex] or [latex](h \u00b1 b, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Foci: [latex](h \u00b1 c, k)[\/latex] or [latex](h, k \u00b1 c)[\/latex], where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.\r\n[latex]4{x}^{2}+{y}^{2}-24x+2y+21=0[\/latex][reveal-answer q=\"141744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"141744\"][latex]\\dfrac{{\\left(x - 3\\right)}^{2}}{4}+\\dfrac{{\\left(y+1\\right)}^{2}}{16}=1[\/latex]; center: [latex]\\left(3,-1\\right)[\/latex]; vertices: [latex]\\left(3,-\\text{5}\\right)[\/latex] and [latex]\\left(3,\\text{3}\\right)[\/latex]; co-vertices: [latex]\\left(1,-1\\right)[\/latex] and [latex]\\left(5,-1\\right)[\/latex]; foci: [latex]\\left(3,-\\text{1}-2\\sqrt{3}\\right)[\/latex] and [latex]\\left(3,-\\text{1+}2\\sqrt{3}\\right)[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write equations of ellipses in standard form.<\/li>\n<li>Graph ellipses.<\/li>\n<li>Solve applied problems involving ellipses.<\/li>\n<\/ul>\n<\/section>\n<h2>Ellipses<\/h2>\n<p>An <strong>ellipse<\/strong> is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a <strong>focus<\/strong> (plural: <strong>foci<\/strong>) of the ellipse.We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03202213\/CNX_Precalc_Figure_10_01_0032.jpg\" alt=\"\" width=\"487\" height=\"560\" \/><figcaption class=\"wp-caption-text\">Visual representation of how to draw an ellipse<\/figcaption><\/figure>\n<p>Every ellipse has two axes of symmetry. The longer axis is called the <strong>major axis<\/strong>, and the shorter axis is called the <strong>minor axis<\/strong>. Each endpoint of the major axis is the <strong>vertex<\/strong> of the ellipse (plural: <strong>vertices<\/strong>), and each endpoint of the minor axis is a <strong>co-vertex<\/strong> of the ellipse. The <strong>center of an ellipse<\/strong> is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci.<\/p>\n<p>To derive the equation of an\u00a0<span class=\"no-emphasis\">ellipse<\/span>\u00a0centered at the origin, we begin with the foci [latex](-c,0)[\/latex] and\u00a0[latex](-c,0)[\/latex]. The ellipse is the set of all points\u00a0[latex](x,y)[\/latex] such that the sum of the distances from\u00a0[latex](x,y)[\/latex] to the foci is constant, as shown in the figure below.<\/p>\n<figure id=\"attachment_2327\" aria-describedby=\"caption-attachment-2327\" style=\"width: 530px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2327\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03202217\/CNX_Precalc_Figure_10_01_0142-300x169.jpg\" alt=\"\" width=\"530\" height=\"299\" \/><figcaption id=\"caption-attachment-2327\" class=\"wp-caption-text\">Ellipse with key features labeled<\/figcaption><\/figure>\n<p id=\"fs-id1226529\">If [latex](a,0)[\/latex] is a\u00a0<span class=\"no-emphasis\">vertex<\/span>\u00a0of the ellipse, the distance from\u00a0[latex](-c,0)[\/latex] to [latex](a,0)[\/latex] is [latex]a-(-c)=a+c[\/latex]. The distance from [latex](c,0)[\/latex] to [latex](a,0)[\/latex] is [latex]a-c[\/latex]. The sum of the distances from the\u00a0<span class=\"no-emphasis\">foci<\/span>\u00a0to the vertex is<\/p>\n<p style=\"text-align: center;\">[latex](a+c)+(a-c)=2a[\/latex]<\/p>\n<p style=\"text-align: left;\">If [latex](x,y)[\/latex] is a point on the ellipse, then we can define the following variables:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}d_1&=\\text{the distance from } (-c,0) \\text{ to } (x,y) \\\\ d_2&= \\text{the distance from } (c,0) \\text{ to } (x,y) \\end{align}[\/latex]<\/p>\n<p style=\"text-align: left;\">By the definition of an ellipse, [latex]d_1+d_2[\/latex] is constant for any point [latex](x,y)[\/latex] on the ellipse. We know that the sum of these distances is [latex]2a[\/latex] for the vertex [latex](a,0)[\/latex]. It follows that [latex]d_1+d_2=2a[\/latex] for any point on the ellipse. The derivation of the standard form of the equation of an ellipse relies on this relationship and the distance formula. The derivation is beyond the scope of this course, but the equation is:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1[\/latex]<\/p>\n<p>for an ellipse centered at the origin with its major axis on the\u00a0[latex]x[\/latex]-axis and<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{x^2}{b^2}+\\dfrac{y^2}{a^2}=1[\/latex]<\/p>\n<p>for an ellipse centered at the origin with its major axis on the [latex]y[\/latex]-axis.<\/p>\n<h2>Writing Equations and Graphing Ellipses Centered at the Origin in Standard Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Standard Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal major axis: [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1[\/latex], where [latex]a > b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical major axis: [latex]\\frac{x^2}{b^2} + \\frac{y^2}{a^2} = 1[\/latex], where [latex]a > b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Important Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center: Always [latex](0, 0)[\/latex] for these forms<\/li>\n<li class=\"whitespace-normal break-words\">Vertices: [latex](\u00b1a, 0)[\/latex] for horizontal, [latex](0, \u00b1a)[\/latex] for vertical<\/li>\n<li class=\"whitespace-normal break-words\">Co-vertices: [latex](0, \u00b1b)[\/latex] for horizontal, [latex](\u00b1b, 0)[\/latex] for vertical<\/li>\n<li class=\"whitespace-normal break-words\">Foci: [latex](\u00b1c, 0)[\/latex] for horizontal, [latex](0, \u00b1c)[\/latex] for vertical, where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Axes Lengths:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Major axis: [latex]2a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Minor axis: [latex]2b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>From Equation to Graph<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify orientation (horizontal\/vertical) based on larger denominator<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]a[\/latex] and [latex]b[\/latex] from equation<\/li>\n<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] using [latex]c^2 = a^2 - b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Plot center, vertices, co-vertices, and foci<\/li>\n<li class=\"whitespace-normal break-words\">Sketch the ellipse through these points<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>From Points to Equation<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determine orientation from given points<\/li>\n<li class=\"whitespace-normal break-words\">Identify [latex]a[\/latex] and [latex]c[\/latex] from vertices and foci<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex] using [latex]b^2 = a^2 - c^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute values into appropriate standard form<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the ellipse that has vertices [latex]\\left(0,\\pm 8\\right)[\/latex] and foci\u00a0[latex](0,\\pm \\sqrt{5})[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q997539\">Show Solution<\/button><\/p>\n<div id=\"q997539\" class=\"hidden-answer\" style=\"display: none\">[latex]x^2+\\dfrac{y^2}{16}=1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the ellipse given by the equation [latex]\\dfrac{{x}^{2}}{36}+\\dfrac{{y}^{2}}{4}=1[\/latex]. Identify and label the center, vertices, co-vertices, and foci.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q796288\">Show Solution<\/button><\/p>\n<div id=\"q796288\" class=\"hidden-answer\" style=\"display: none\">center: [latex]\\left(0,0\\right)[\/latex]; vertices: [latex]\\left(\\pm 6,0\\right)[\/latex]; co-vertices: [latex]\\left(0,\\pm 2\\right)[\/latex]; foci: [latex]\\left(\\pm 4\\sqrt{2},0\\right)[\/latex]<\/p>\n<figure id=\"attachment_3258\" aria-describedby=\"caption-attachment-3258\" style=\"width: 731px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02000914\/CNX_Precalc_Figure_10_01_0082.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3258 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02000914\/CNX_Precalc_Figure_10_01_0082.jpg\" alt=\"\" width=\"731\" height=\"366\" \/><\/a><figcaption id=\"caption-attachment-3258\" class=\"wp-caption-text\">Ellipse with the center and key points labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the ellipse given by the equation [latex]49{x}^{2}+16{y}^{2}=784[\/latex]. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q866773\">Show Solution<\/button><\/p>\n<div id=\"q866773\" class=\"hidden-answer\" style=\"display: none\">Standard form: [latex]\\dfrac{{x}^{2}}{16}+\\dfrac{{y}^{2}}{49}=1[\/latex]; center: [latex]\\left(0,0\\right)[\/latex]; vertices: [latex]\\left(0,\\pm 7\\right)[\/latex]; co-vertices: [latex]\\left(\\pm 4,0\\right)[\/latex]; foci: [latex]\\left(0,\\pm \\sqrt{33}\\right)[\/latex]<\/p>\n<figure id=\"attachment_3259\" aria-describedby=\"caption-attachment-3259\" style=\"width: 731px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02001223\/CNX_Precalc_Figure_10_01_0102.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3259 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02001223\/CNX_Precalc_Figure_10_01_0102.jpg\" alt=\"\" width=\"731\" height=\"741\" \/><\/a><figcaption id=\"caption-attachment-3259\" class=\"wp-caption-text\">Ellipse with the center and key points labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dbcdgabg-oZB69DY0q9A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/oZB69DY0q9A?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dbcdgabg-oZB69DY0q9A\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851181&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dbcdgabg-oZB69DY0q9A&amp;vembed=0&amp;video_id=oZB69DY0q9A&amp;video_target=tpm-plugin-dbcdgabg-oZB69DY0q9A\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Conic+Sections+-++The+Ellipse+part+2+of+2_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConic Sections: The Ellipse part 2 of 2\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-chehfegb-azI5kALyiXs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/azI5kALyiXs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-chehfegb-azI5kALyiXs\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851279&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-chehfegb-azI5kALyiXs&amp;vembed=0&amp;video_id=azI5kALyiXs&amp;video_target=tpm-plugin-chehfegb-azI5kALyiXs\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Graph+an+Ellipse+with+Center+at+the+Origin+and+Horizontal+Major+Axis_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Graph an Ellipse with Center at the Origin and Horizontal Major Axis\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Writing Equations and Graphing Ellipses Not Centered at the Origin<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Standard Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal major axis: [latex]\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1[\/latex], where [latex]a > b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical major axis: [latex]\\frac{(x-h)^2}{b^2} + \\frac{(y-k)^2}{a^2} = 1[\/latex], where [latex]a > b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center: [latex](h, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertices: [latex](h \u00b1 a, k)[\/latex] for horizontal, [latex](h, k \u00b1 a)[\/latex] for vertical<\/li>\n<li class=\"whitespace-normal break-words\">Co-vertices: [latex](h, k \u00b1 b)[\/latex] for horizontal, [latex](h \u00b1 b, k)[\/latex] for vertical<\/li>\n<li class=\"whitespace-normal break-words\">Foci: [latex](h \u00b1 c, k)[\/latex] for horizontal, [latex](h, k \u00b1 c)[\/latex] for vertical, where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Axes Lengths:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Major axis: [latex]2a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Minor axis: [latex]2b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>From Equation to Graph<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify orientation (horizontal\/vertical) based on larger denominator<\/li>\n<li class=\"whitespace-normal break-words\">Determine center (h, k) from equation<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]a[\/latex] and [latex]b[\/latex] from equation<\/li>\n<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] using [latex]c^2 = a^2 - b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Plot center, vertices, co-vertices, and foci<\/li>\n<li class=\"whitespace-normal break-words\">Sketch the ellipse through these points<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>From Points to Equation<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Determine orientation from given points<\/li>\n<li class=\"whitespace-normal break-words\">Find center using midpoint formula<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]a[\/latex] (half the distance between vertices)<\/li>\n<li class=\"whitespace-normal break-words\">Find [latex]c[\/latex] (distance from center to focus)<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]b[\/latex] using [latex]b^2 = a^2 - c^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute values into appropriate standard form<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">What is the standard form equation of the ellipse that has vertices [latex]\\left(-3,3\\right)[\/latex] and [latex]\\left(5,3\\right)[\/latex] and foci [latex]\\left(1 - 2\\sqrt{3},3\\right)[\/latex] and [latex]\\left(1+2\\sqrt{3},3\\right)?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q256644\">Show Solution<\/button><\/p>\n<div id=\"q256644\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{{\\left(x - 1\\right)}^{2}}{16}+\\dfrac{{\\left(y - 3\\right)}^{2}}{4}=1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the ellipse given by the equation [latex]\\dfrac{{\\left(x - 4\\right)}^{2}}{36}+\\dfrac{{\\left(y - 2\\right)}^{2}}{20}=1[\/latex]. Identify and label the center, vertices, co-vertices, and foci.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q590423\">Show Solution<\/button><\/p>\n<div id=\"q590423\" class=\"hidden-answer\" style=\"display: none\">Center: [latex]\\left(4,2\\right)[\/latex]; vertices: [latex]\\left(-2,2\\right)[\/latex] and [latex]\\left(10,2\\right)[\/latex]; co-vertices: [latex]\\left(4,2 - 2\\sqrt{5}\\right)[\/latex] and [latex]\\left(4,2+2\\sqrt{5}\\right)[\/latex]; foci: [latex]\\left(0,2\\right)[\/latex] and [latex]\\left(8,2\\right)[\/latex]<\/p>\n<figure id=\"attachment_3260\" aria-describedby=\"caption-attachment-3260\" style=\"width: 731px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02001521\/CNX_Precalc_Figure_10_01_0122.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3260 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02001521\/CNX_Precalc_Figure_10_01_0122.jpg\" alt=\"\" width=\"731\" height=\"666\" \/><\/a><figcaption id=\"caption-attachment-3260\" class=\"wp-caption-text\">Ellipse with the center and key points labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gchbachg-tJmp1PJD9o8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tJmp1PJD9o8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gchbachg-tJmp1PJD9o8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851280&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-gchbachg-tJmp1PJD9o8&amp;vembed=0&amp;video_id=tJmp1PJD9o8&amp;video_target=tpm-plugin-gchbachg-tJmp1PJD9o8\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Graph+an+Ellipse+with+Center+NOT+at+the+Origin+and+Horizontal+Major+Axis_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Graph an Ellipse with Center NOT at the Origin and Horizontal Major Axis\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Graphing an Ellipse in General Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">General Form: [latex]ax^2 + by^2 + cx + dy + e = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Goal: Transform to Standard Form [latex]\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Transformation Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Group like terms<\/li>\n<li class=\"whitespace-normal break-words\">Complete the square for [latex]x[\/latex] and [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Divide by constant term<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Features:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center [latex](h, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertices: [latex](h \u00b1 a, k)[\/latex] or [latex](h, k \u00b1 a)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Co-vertices: [latex](h, k \u00b1 b)[\/latex] or [latex](h \u00b1 b, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Foci: [latex](h \u00b1 c, k)[\/latex] or [latex](h, k \u00b1 c)[\/latex], where [latex]c^2 = a^2 - b^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.<br \/>\n[latex]4{x}^{2}+{y}^{2}-24x+2y+21=0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q141744\">Show Solution<\/button><\/p>\n<div id=\"q141744\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{{\\left(x - 3\\right)}^{2}}{4}+\\dfrac{{\\left(y+1\\right)}^{2}}{16}=1[\/latex]; center: [latex]\\left(3,-1\\right)[\/latex]; vertices: [latex]\\left(3,-\\text{5}\\right)[\/latex] and [latex]\\left(3,\\text{3}\\right)[\/latex]; co-vertices: [latex]\\left(1,-1\\right)[\/latex] and [latex]\\left(5,-1\\right)[\/latex]; foci: [latex]\\left(3,-\\text{1}-2\\sqrt{3}\\right)[\/latex] and [latex]\\left(3,-\\text{1+}2\\sqrt{3}\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Conic Sections: The Ellipse part 2 of 2\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/oZB69DY0q9A\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Graph an Ellipse with Center at the Origin and Horizontal Major Axis\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/azI5kALyiXs?si=8XsWXcAhZFUcMrZi\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Graph an Ellipse with Center NOT at the Origin and Horizontal Major Axis\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/tJmp1PJD9o8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Conic Sections: The Ellipse part 2 of 2","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/oZB69DY0q9A","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 1: Graph an Ellipse with Center at the Origin and Horizontal Major Axis","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/azI5kALyiXs?si=8XsWXcAhZFUcMrZi","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 1: Graph an Ellipse with Center NOT at the Origin and Horizontal Major Axis","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/tJmp1PJD9o8","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851181&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dbcdgabg-oZB69DY0q9A&vembed=0&video_id=oZB69DY0q9A&video_target=tpm-plugin-dbcdgabg-oZB69DY0q9A'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851279&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-chehfegb-azI5kALyiXs&vembed=0&video_id=azI5kALyiXs&video_target=tpm-plugin-chehfegb-azI5kALyiXs'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851280&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gchbachg-tJmp1PJD9o8&vembed=0&video_id=tJmp1PJD9o8&video_target=tpm-plugin-gchbachg-tJmp1PJD9o8'><\/script>\n","media_targets":["tpm-plugin-dbcdgabg-oZB69DY0q9A","tpm-plugin-chehfegb-azI5kALyiXs","tpm-plugin-gchbachg-tJmp1PJD9o8"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1591"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1591\/revisions"}],"predecessor-version":[{"id":5639,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1591\/revisions\/5639"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1591\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1591"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1591"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1591"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1591"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}