# Equation of an Ellipse

## Ellipses Centered at (0, 0)

### Horizontal Major Axis

**F'(-c, 0) F(c, 0)**

### Vertical Major Axis

**F'(0, -c) F(0, c)**

## Ellipses Centered at (x_{0}, y_{0})

### Horizontal Major Axis

By removing the denominators, an equation of the following form is obtained:

A and B have the same sign.

**F'(x _{0}−c, y_{0}) F(x_{0}+c, y_{0})**

### Vertical Major Axis

F'(x_{0}, y_{0}−c) F(x_{0}, y+c)

### Examples

Find the elements and the equation of the ellipse with foci: F' = (−3, 0), F = (3, 0) and a major axis of 10.

Given the equation of the ellipse , determine the eccentricity and find the coordinates of the vertices and foci.

Find the equation of the ellipse that has a focus of F = (7, 2), vertex of A = (9, 2) and a center of C = (4, 2).

Given the ellipse equation , determine the center, semiaxes, vertices and foci.

Determine and plot the coordinates of the foci, vertices and eccentricity of the following ellipses:

1

2

3

4