# Equation of an Ellipse

An ellipse is the set of all points in a plane such that the sum of the distances from T to two fixed points F_{1} and F_{2} is a given constant, K.

In the demonstration below, F_{1} and F_{2} are the two blue thumb tacks, and the the fixed distance is the length of the rope.

TF_{1} + TF_{2} = KF_{1} and F_{2} are both foci(plural of focus) of the ellipse.

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 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.

Thevertices 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.

You can think of an ellipse as an oval.

### **Picture** of an Ellipse

### **Standard Form Equation** of an Ellipse

The general form for the standard form equation of an ellipse is

Horizontal Major Axis Example

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

- 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 Cartesian plane

- 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

### **Practice** Problem

Can you determine the values of a and b for the equation of the ellipse pictured below?

Can you determine the values of a and b for the equation of the ellipse pictured in the graph below?

What are values of a and b for the standard form equation

of the ellipse in the graph?

Examine the graph of the ellipse below to determine a and b for the standard form equation?

Examine the graph of the ellipse below to determine a and b for the standard form equation?

What is the standard form equation of the ellipse in the graph below?

What is the standard form equation of the ellipse in the graph below?

### 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.Can you graph the equation of the ellipse below and find the values of a and b?

Can you graph the equation of the ellipse below ? What are the values of a and b?

Can you graph the equation of the ellipse below? What are the values of a and b?

Determine the values of a and b as well as what the graph of the ellipse with the equation shown below.

Determine the values of a and b as well as what the graph of the ellipse with the equation shown below.

Determine the values of a and b as well as what the graph of the ellipse with the equation shown below.

- Worksheet Version of this Web page (same questions on a worksheet)

Category: Writing