# Summary: The Ellipse

## Key Equations

Horizontal ellipse, center at origin | [latex]\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex] |

Vertical ellipse, center at origin | [latex]\dfrac{{x}^{2}}{{b}^{2}}+\dfrac{{y}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex] |

Horizontal ellipse, center [latex]\left(h,k\right)[/latex] | [latex]\dfrac{{\left(x-h\right)}^{2}}{{a}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex] |

Vertical ellipse, center [latex]\left(h,k\right)[/latex] | [latex]\dfrac{{\left(x-h\right)}^{2}}{{b}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex] |

## Key Concepts

- An ellipse 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 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 [latex]\left(x,y\right)[/latex] 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 [latex]\left(x,y\right)[/latex] 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 ellipse

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