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how to create an ellipse

Equation of an Ellipse

Standard Form equation

Vertices and Axis

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

major axes of ellipse
minor axes of ellipse

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.
standard form equation ellipse

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.
standard form equation ellipse
Picture of an 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 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..

equation of ellipse
equation of ellipse

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

equation of ellipse x term highlighted
equation of ellipse x term highlighted

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

equation of ellipse y term highlighted
equation of ellipse y term highlighted

Practice Problem

Problem 1

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 2

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

More Practice writing equation from the Graph

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 ellipse equation

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?

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

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

practice
Problem 9

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

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

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

practice
Problem 10

Can you graph the ellipse with the equation below?

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

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

Practice
Problem 11

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

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

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

Practice
Problem 12

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

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

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

Practice
Problem 13

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

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

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

Practice
Problem 14

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

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

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

Practice
Problem 15

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

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

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

Practice
Problem 16

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

$ 9x^2 + 4y^2 = 36 $

Equation

Here is a picture of the ellipse's graph.

Practice

Back to Conic Sections Next to Focus of Ellipse