Equation of an Ellipse

Standard Form equation

An ellipse is the set of all points in a plane such that the sum of the distances from T to two fixed points F1 and F2 is a given constant, K.

                    In the demonstration below, F1 and F2 are the two blue thumb tacks, and the the fixed distance is the length of the rope.
                    
                    

How to Create an Ellipse Demonstration

how to create an ellipse

TF1 + TF2 = K F1 and F2 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.

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.


You can think of an ellipse as an oval.

strong>Picture of an Ellipse

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

Problem 1

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

Practice Problem equation of ellipse
Problem 2

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

Practice Problem equation of ellipse
Problem 3

What are values of a and b for the standard form equation of the ellipse in the graph?

Practice Problem equation of ellipse
Problem 4

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

Practice Problem equation of ellipse
Problem 5

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

Practice Problem equation of ellipse
Problem 6

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

Practice Problem equation of ellipse
Problem 7

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

Practice Problem equation of ellipse
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

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

  • a = 5
  • b = 2
Problem 9

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

  • a = 5
  • b = 3
Problem 10

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

  • a = 6
  • b = 5
Problem 11

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

  • a = 6
  • b = 5
Problem 12

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

  • a = 6
  • b = 2
Problem 13

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

  • a = 6
  • b = 1
Problem 14

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

  • a = 6
  • b = 2
Problem 15

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

  • a = 7
  • b = 3
Problem 16

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

  • a = 3
  • b = 2
Here is a picture of the ellipse's graph.