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## Precalculus (2018 edition)

### Course: Precalculus (2018 edition)>Unit 2

Lesson 6: Foci of an ellipse

# Ellipse foci review

Review your knowledge of the foci of an ellipse.

## What are the foci of an ellipse?

The $\text{foci}$ 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 $\text{major radius}$.
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 $q$:
${f}^{2}={p}^{2}-{q}^{2}$

## Finding the foci of an ellipse

Given the radii of an ellipse, we can use the equation ${f}^{2}={p}^{2}-{q}^{2}$ 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.
$\begin{array}{rl}{f}^{2}& ={p}^{2}-{q}^{2}\\ \\ {f}^{2}& ={5}^{2}-{4}^{2}\\ \\ {f}^{2}& =9\\ \\ f& =3\end{array}$
The major axis is the horizontal one, so the foci lie $3$ units to the right and left of the center. In other words, the foci lie at $\left(-4±3,3\right)$, which are $\left(-7,3\right)$ and $\left(-1,3\right)$.