Summary: The Ellipse Key Equations Horizontal ellipse, center at origin x2a2+y2b2=1, a>b\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1,\text{ }a>ba2x2+b2y2=1, a>b Vertical ellipse, center at origin x2b2+y2a2=1, a>b\dfrac{{x}^{2}}{{b}^{2}}+\dfrac{{y}^{2}}{{a}^{2}}=1,\text{ }a>bb2x2+a2y2=1, a>b Horizontal ellipse, center (h,k)\left(h,k\right)(h,k) (x−h)2a2+(y−k)2b2=1, a>b\dfrac{{\left(x-h\right)}^{2}}{{a}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{ }a>ba2(x−h)2+b2(y−k)2=1, a>b Vertical ellipse, center (h,k)\left(h,k\right)(h,k) (x−h)2b2+(y−k)2a2=1, a>b\dfrac{{\left(x-h\right)}^{2}}{{b}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{a}^{2}}=1,\text{ }a>bb2(x−h)2+a2(y−k)2=1, a>b Key Concepts An ellipse is the set of all points (x,y)\left(x,y\right)(x,y) in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci). When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. Glossary center of an ellipse the midpoint of both the major and minor axes conic section any shape resulting from the intersection of a right circular cone with a plane ellipse the set of all points (x,y)\left(x,y\right)(x,y) in a plane such that the sum of their distances from two fixed points is a constant foci plural of focus focus (of an ellipse) one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point (x,y)\left(x,y\right)(x,y) on the ellipse is a constant major axis the longer of the two axes of an ellipse minor axis the shorter of the two axes of an ellipseLicenses & AttributionsCC licensed content, OriginalRevision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.CC licensed content, Shared previouslyCollege Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2.CC licensed content, Specific attributionPrecalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.
