# Equation of an Ellipse

Standard Form equation

### Vertices and Axis

Before looking at the ellispe equation below, you should know a few terms.

More Examples of Axes, Vertices, Co-vertices

### Horizontal Major Axis Example

Example of the graph and equation of an ellipse on the

• The major axis of this ellipse is horizontal and is the red segment from (-2, 0) to (2, 0).
• The center of this ellipse is the origin since (0, 0) is the midpoint of the major axis.
• The value of a = 2 and b = 1.

### Vertical Major Axis Example

Example of the graph and equation of an ellipse on the :

• The major axis of this ellipse is vertical and is the red segment from (2, 0) to (-2, 0).
• The center of this ellipse is the origin since (0, 0) is the midpoint of the major axis.
• The value of a = 2 and b = 1.

The major axis is the segment that contains both foci and has its endpoints on the ellipse. These endpoints are called the vertices. The midpoint of the major axis is the center of the ellipse.

The minor axis is perpendicular to the major axis at the center, and the endpoints of the minor axis are called co-vertices.

The vertices are at the intersection of the major axis and the ellipse.

The co-vertices are at the intersection of the minor axis and the ellipse.

### Standard Form Equation of an Ellipse

The general form for the standard form equation of an ellipse is shown below..

In the equation, the denominator under the $$x^2$$ term is the square of the x coordinate at the x -axis.

The denominator under the $$y^2$$ term is the square of the y coordinate at the y-axis.

### Practice Problem

##### Problem 3

More Practice writing equation from the Graph

##### Problem 7
Since a = b in the ellipse below, this ellipse is actually a circle whose standard form equation is x² + y² = 9

### Graph of Ellipse from the Equation

The problems below provide practice creating the graph of an ellipse from the equation of the ellipse. All practice problems on this page have the ellipse centered at the origin.

Click here for practice problems involving an ellipse not centered at the origin.
##### Problem 8

$\frac {x^2}{\red 2^2} + \frac{y^2}{\red 5^2} = 1$

##### Problem 9

$\frac {x^2}{25} + \frac{y^2}{9} = 1 \\ \frac {x^2}{\red 5^2} + \frac{y^2}{\red 3^2} = 1 \\$

##### Problem 10

$\frac {x^2}{25} + \frac{y^2}{36} = 1 \\ \frac {x^2}{\red 5^2} + \frac{y^2}{\red 6^2} = 1 \\$

##### Problem 11

$\frac {x^2}{36} + \frac{y^2}{25} = 1 \\ \frac {x^2}{\red 6^2} + \frac{y^2}{\red 5^2} = 1 \\$

##### Problem 12

$\frac {x^2}{36} + \frac{y^2}{4} = 1 \\ \frac {x^2}{\red 6^2} + \frac{y^2}{\red 2^2} = 1 \\$

##### Problem 13

$\frac {x^2}{1} + \frac{y^2}{36} = 1 \\ \frac {x^2}{\red 1^2} + \frac{y^2}{\red 6^2} = 1 \\$

##### Problem 14

$\frac {x^2}{36} + \frac{y^2}{4} = 1 \\ \frac {x^2}{\red 6^2} + \frac{y^2}{\red 2^2} = 1 \\$

##### Problem 15

$\frac {x^2}{36} + \frac{y^2}{9} = 1 \\ \frac {x^2}{\red 6^2} + \frac{y^2}{\red 3^2} = 1$

##### Problem 16

Here is a picture of the ellipse's graph.