>Ellipses>What Is the Equation of an Ellipse?

What Is the Equation of an Ellipse?

Ellipse with focal point and axes

Theory

The Ellipse

An ellipse consists of all points that have the same total distance to two given points, called focal points, or foci.

The following distances are important to describe an ellipse:

Semi-major axis ( $a$ ):

The greatest distance from the center of the ellipse to any point on the ellipse.

Semi-minor axis ( $b$ ):

The shortest distance from the center of the ellipse to any point on the ellipse.

Rule

The Standard Equation of an Ellipse

If both the focal points of an ellipse lie on the $x$ -axis equidistant to the origin, and you call the semi-major axis $a$ and the semi-minor axis $b$ , the standard equation for the ellipse is

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

A circle is a special case of an ellipse, where the focal points coincide and the semi-major and semi-minor axes are equal.

Example 1

Find the standard equation of an ellipse with a semi-major axis of $a = 5$ and a semi-minor axis of $b = 3$

\begin{array}{l} x 2 5^{2} + \frac{y^{2}}{3^{2}} = 1 \frac{x^{2}}{25} + \frac{y^{2}}{9} & = 1Example 2 Show that 9 x^{2} + 4 y^{2} - 36 = 0 is the equation of an ellipse To show that this equation represents an ellipse, you need to change it into the form of the standard equation of an ellipse. You can do that like this: \begin{array}{l} 9 x^{2} + 4 y^{2} - 36 & = 0 \\ 9 x^{2} + 4 y^{2} & = 36 & | \div 9 \\ x^{2} + \frac{4 y^{2}}{9} & = 4 & | \div 4 \\ \frac{x^{2}}{4} + \frac{y^{2}}{9} & = 1 \end{array} Because you’ve been able to change the expression into the form of the standard equation of an ellipse, you’ve shown that 9 x^{2} + 4 y^{2} - 36 = 0 is an ellipse with semi-minor axis a = \sqrt{4} = 2 and semi-major axis b = \sqrt{9} = 3 .Want to know more? Sign Up It's free!Previous entry What Are Ellipses?Go backHome All Math Games Progress SpaceAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com Booking of private tutoring +47 22 150 300 Address Bygdøy allé 23
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            Norway FAQ Links WeTakeAction Math Magic for Ukraine Tutoring Bootcamps Junior Math Pen & Paper exercises Encyclopedia Book a Tutor Language Norsk Українська Terms Privacy{"props":{"pageProps":{"campaign":false,"topic":{"cssUrl":"/0e751716-e4cf-4658-ac87-c97d9167170e-en-1.css","breadcrumbs":[{"h1":"Encyclopedia","_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c","path":"/encyclopedia"},{"path":"/encyclopedia/geometry","h1":"Geometry","_key":"20426156-ae03-5b57-a367-dd3f5184f133"},{"path":"/encyclopedia/geometry/planar-figures","h1":"Planar Figures","_key":"cbc8f010-4795-5801-97ad-13b89d47cb61"},{"path":"/encyclopedia/geometry/planar-figures/circles","h1":"Circles","_key":"1d77cfae-543b-5850-8157-34eb8da4c8cb"},{"path":"/encyclopedia/geometry/planar-figures/circles/ellipses","h1":"Ellipses","_key":"d75d095f-0a86-5957-80e7-a6037db46c19"},{"path":"/encyclopedia/geometry/planar-figures/circles/ellipses/what-is-the-equation-of-an-ellipse","h1":"What Is the Equation of an Ellipse?","_key":"d8cc679e-e0d1-57d9-86f3-c91ca8f95818"}],"taskReference":null,"isRootPath":false,"sharedLanguageKey":null,"type":"encyclopedia","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     \u003cdiv class=\"tight\"\u003e  \u003c/p\u003e \u003cdiv class=\"center\"  \u003e                                                                                                                                                                                                                                          \u003cp class=\"noindent\" \u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/ellipseeng.svg\" alt=\"Ellipse with focal point and axes\" width=\"572\" height=\"572\"\u003e\u003c/p\u003e\u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/div\u003e\u003cdiv class=\"wide\"\u003e\u003c/p\u003e \u003cdiv class=\"center\"  \u003e \u003cp class=\"noindent\" \u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/ellipseeng.svg\" alt=\"Ellipse with focal point and axes\" width=\"451\" height=\"451\"\u003e\u003c/p\u003e\u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/div\u003e \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooinfo\" id=\"tcolobox-847\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Theory\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"info-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eThe\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eEllipse\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eAn \u003ca href=\"/encyclopedia/geometry/planar-figures/circles/ellipses\"\u003eellipse\u003c/a\u003e consists of all points that have the same total distance to two given points, called \u003cspan  class=\"ec-lmri-12x-x-144\"\u003efocal\u003c/span\u003e \u003cspan  class=\"ec-lmri-12x-x-144\"\u003epoints\u003c/span\u003e, or foci. \u003c/p\u003e\u003cp class=\"noindent\" \u003eThe following distances are important to describe an ellipse:         \u003c/p\u003e\u003cdl class=\"description\"\u003e\u003cdt class=\"description\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSemi-major axis (\u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003e):\u003c/span\u003e \u003c/dt\u003e\u003cdd  class=\"description\"\u003e       \u003cp class=\"noindent\" \u003eThe         greatest         distance         from         the         center         of         the         ellipse         to         any         point         on         the         ellipse.         \u003c/p\u003e\u003c/dd\u003e\u003cdt class=\"description\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSemi-minor axis (\u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/math\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003e):\u003c/span\u003e \u003c/dt\u003e\u003cdd  class=\"description\"\u003e       \u003cp class=\"noindent\" \u003eThe         shortest         distance         from         the         center         of         the         ellipse         to         any         point         on         the         ellipse. \u003c/p\u003e         \u003cp class=\"noindent\" \u003e\u003c/dd\u003e\u003c/dl\u003e \u003c/p\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-848\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Rule\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eThe\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eStandard\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eEquation\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eof\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ean\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eEllipse\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eIf both the focal points of an ellipse lie on the \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/math\u003e-axis equidistant to the \u003ca href=\"/encyclopedia/functions/theory-of-functions/fundamentals-of-functions/what-is-a-coordinate-system\"\u003eorigin\u003c/a\u003e, and you call the semi-major axis \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and the semi-minor axis \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/math\u003e, the standard \u003ca href=\"/encyclopedia/algebra/equations-and-inequalities/equations/linear-equations/what-is-an-equation\"\u003eequation\u003c/a\u003e for the ellipse is \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                                \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                              \u003cp class=\"indent\" \u003e     A \u003ca href=\"/encyclopedia/geometry/planar-figures/circles/about-the-circle/what-is-the-circumference-and-area-of-a-circle\"\u003ecircle\u003c/a\u003e is a special case of an ellipse, where the focal points coincide and the semi-major and semi-minor axes are equal. \u003c/p\u003e \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-849\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eFind\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ethe\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003estandard\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eequation\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eof\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ean\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eellipse\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ewith\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ea\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003esemi-major\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eaxis\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eof\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eand\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ea\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003esemi-minor\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eaxis\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eof\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eb\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" mathvariant=\"bold-italic\"\u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"align-star\"\u003e                                     \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e \u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/mtd\u003e                                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e \u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e    \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                           \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-850\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Example 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eShow\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ethat\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eis\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ethe\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eequation\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eof\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ean\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eellipse\u003c/span\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003eTo show that this equation represents an ellipse, you need to change it into the form of the standard equation of an ellipse. You can do that like this: \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"alignat-star\"\u003e                                     \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e÷\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e    \u003cmrow  \u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003c/mtd\u003e                                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"1em\" class=\"quad\"/\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e÷\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e                                     \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003c/mtd\u003e                                             \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003eBecause you’ve been able to change the expression into the form of the standard equation of an ellipse, you’ve shown that \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e is an ellipse with semi-minor axis \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/math\u003e and semi-major axis \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eb\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmsqrt\u003e\u003cmrow\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003c/msqrt\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e                                                                                                                            \u003cp class=\"indent\" \u003e     "},"image":{"dimension":"width","value":0.95,"url":"https://images.houseofmath.com/overleaf-assets/images/ellipseeng.svg","alt":"Ellipse with focal point and axes","_key":"03d87971-51a1-5976-96e3-12da29d60e82","type":"textwidth"},"siblings":[{"h1":"What Are Ellipses?","path":"/encyclopedia/geometry/planar-figures/circles/ellipses/what-are-ellipses"},{"h1":"What Is the Equation of an Ellipse?","path":"/encyclopedia/geometry/planar-figures/circles/ellipses/what-is-the-equation-of-an-ellipse"}],"otherLocales":[],"h1":"What Is the Equation of an Ellipse?","metaDescription":"Learn about the properties of the ellipse, the standard equation of the ellipse and how you can find the equation of the ellipse by reading this entry.","locale":"en","language":"en","previewUrl":"https://images.houseofmath.com/eyJrZXkiOiJvdmVybGVhZi1hc3NldHMvcHJldmlldy8zMDJlcXVhdGlvbmVsbGlwc2UucG5nIiwiYnVja2V0IjoiaG9tLWZyb250ZW5kLWFzc2V0cyIsInZlcnNpb24iOjN9","key":"content/en/encyclopedia/03 Geometry/01 Planar Figures/04 Circles/03 Ellipses/302EquationEllipse.tex","path":"/encyclopedia/geometry/planar-figures/circles/ellipses/what-is-the-equation-of-an-ellipse","title":"What Is the Equation of an Ellipse?"},"topicOverview":[{"h1":"Numbers and Quantities","path":"/encyclopedia/numbers-and-quantities","shouldRenderChildren":true},{"h1":"Algebra","path":"/encyclopedia/algebra","shouldRenderChildren":true},{"h1":"Geometry","path":"/encyclopedia/geometry","shouldRenderChildren":true,"children":[{"h1":"Planar 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