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Ellipse foci review

Review your knowledge of the foci of an ellipse.

What are the foci of an ellipse?

The start color #ed5fa6, start text, f, o, c, i, end text, end color #ed5fa6 of an ellipse are two points whose sum of distances from any point on the ellipse is always the same. They lie on the ellipse's start color #1fab54, start text, m, a, j, o, r, space, r, a, d, i, u, s, end text, end color #1fab54.
The distance between each focus and the center is called the focal length of the ellipse. The following equation relates the focal length f with the major radius p and the minor radius q:
f, squared, equals, p, squared, minus, q, squared
Want to learn more about the foci of an ellipse? Check out this video.

Finding the foci of an ellipse

Given the radii of an ellipse, we can use the equation f, squared, equals, p, squared, minus, q, squared to find its focal length. Then, the foci will lie on the major axis, f units away from the center (in each direction). Let's find, for example, the foci of this ellipse:
We can see that the major radius of our ellipse is 5 units, and its minor radius is 4 units.
f2=p2q2f2=5242f2=9f=3\begin{aligned} f^2&=p^2-q^2 \\\\ f^2&=5^2-4^2 \\\\ f^2&=9 \\\\ f&=3 \end{aligned}
The major axis is the horizontal one, so the foci lie start color #1fab54, 3, end color #1fab54 units to the right and left of the center. In other words, the foci lie at left parenthesis, minus, 4, plus minus, start color #1fab54, 3, end color #1fab54, comma, 3, right parenthesis, which are left parenthesis, minus, 7, comma, 3, right parenthesis and left parenthesis, minus, 1, comma, 3, right parenthesis.

Check your understanding

Problem 1
  • Current
Plot the foci of this ellipse.

Want to try more problems like this? Check out this exercise and this exercise.

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