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Ellipse: Definition, Equation & Examples

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  • 0:03 What is an Ellipse?
  • 1:02 The Ellipse Formula
  • 2:51 Example Problems
  • 5:05 Lesson Summary
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Lesson Transcript
Instructor: Joshua White

Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

This lesson will cover the definition of ellipses and the standard form equation of an ellipse. It will also examine how to determine the orientation of an ellipse and how to graph them.

What Is an Ellipse?

You're probably very familiar with circles. They're completely round and are technically defined as all points that are a given distance, known as the radius, from a defined point called the center. But what if you took a circle and stretched it in opposite directions, either left and right or up and down? You'd end up with a shape that is still round, but is obviously no longer a circle. What you actually have now is an ellipse. But what is an ellipse, and how does it work?

An ellipse is defined as the set of all points where the sum of the distances from two fixed points is constant. In other words, there are two fixed points, called foci (or the plural of focus). If you trace out all points where the distance from the first focus to a point plus the distance from the second focus to the same point remains constant, you will draw an ellipse. As mentioned, it's essentially a circle where the circle is stretched vertically or horizontally by equal amounts. Here are two examples:

example of ellipse-1

example of ellipse-2

The Ellipse Formula

Here is the formula for an ellipse in standard form:

ellipse standard form

A^2, b^2, h, and k are all numbers that determine various characteristics about the ellipse. Note the formula has minus signs in it in front of h and k.

From the ellipse formula, you can determine the following items about an ellipse:

  • Orientation
  • Center
  • Height
  • Width
  • Where the foci are located

To determine the orientation, you will compare a^2 and b^2. If a^2 > b^2 (or if the bigger number is under the x), then it will be horizontal, or wider than it is taller. If a^2 < b^2, then you have a vertical ellipse whose height is greater than its width.

The center of the ellipse will be a point (h, k).

To determine the width and height of the ellipse, you'll first find the values of a and b by taking the square root of a^2 and b^2. The width will equal 2a and the height will equal 2b.

Finally, to find the two focus points, or foci, you will need to find the value of c. Note that c isn't given in the formula, but must be found by first finding the value of c^2 by subtracting a^2 and b^2, the two numbers in the denominators of the formula. You will always subtract the smaller value from the larger value; in other words, c^2 should always be a positive number. Then you will take the square root of c^2 to get c, which tells you how far away from the center, either up and down or left and right, the foci are located.

Now, let's take a look at some example problems, including how to graph an ellipse.

Example Problems

Here is the equation of an ellipse:

ellipse example problem 1 formula

Identify the type of ellipse and then graph the ellipse.

This ellipse will be horizontal because the number underneath the x (25) is larger than the number underneath the y (16). The center of the ellipse will be (-2, 0) because h = -2 and k = 0.

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