Ellipses

Ellipses are a rare but important shape to know about for the ACT math section. Here are the main properties of an ellipse that students will be expected to know:

Key Skills for Ellipses

What is an Ellipse?

An ellipse is an oval shape. It is similar to a circle, but where one of its axises is stretched. As a result, an ellipse has both a major axis and a minor axis. 

• The major axis is the axis in between the longer ends of the oval
• The minor axis is the axis in between the shorter ends of the oval

Reading and Graphing Ellipse Equations

Since the shape of an ellipse is similar to a circle, the equation is similar too. An ellipse equation will look like this:

(x–1)29+(y+2)21=1

• Unlike the circle equation, there are now denominators under the squared binomials. This is to differentiate the lengths of the major and minor axes.
• The ellipse equation will always be = 1

Like with the circle equation, the constants inside of the binomial indicate where the center of the ellipse is.

• (x-1)2 means that the center point lies along the vertical line x=1
• (y+2)2means that the center point lies along the horizontal line y= -2
• Combining these two things means the center is at (1,-2)

The denominator term is then equal to half the length of the major/minor axis squared.

• The denominator under the x term is 9, meaning that the each half of the major axis has a length of 3
• The x-axis is the major axis because it has a larger denominator, and therefore its axis is longer
• The denominator under the y term is 1, meaning that each half of the minor axis has a length of 1

ACT questions about ellipses may give an ellipse equation and require students to graph it to find a point on the ellipse or the center point of the ellipse. A question could also provide a graph of an ellipse and then provide several equations, from which the student must then select the equation that correctly matches the graph.