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## Algebra (all content)

### Course: Algebra (all content)>Unit 17

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 $\left(-1,3\right)$.

Problem 1
Plot the foci of this ellipse.

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

## Want to join the conversation?

• Are co-vertexes just the y-axis minor or major radii? Or is it always the minor radii either x or y-axis?
• co-vertices are always the endpoints of the minor vertex, regardless of whether it's parallel to, or on the x-axis or the y-axis
• Does the sum of the two distances from a point to its focus always equal 2*major radius, or can it sometimes equal something else?
• Yes, they always equals twice the major radius. Think about this: if you take a piece of wood, and draw a line on it, then stick a nail at each of the line. Now take a piece of string, and tie each end of the string to one of the nails. The string needs to be longer than the line you drew, so that it's loose with no tension. Then you can take a pen/pencil, use it to push up against the string, and then the string can act as a guide for the pen, and you can trace out an ellipse. The two nails are the foci, and the length of the string is 2*major radius: it's a constant, and never changes.
• Why aren't there lessons for finding the latera recta and the directrices of an ellipse?
• How was the foci discovered?
• The first mention of "foci" was in the multivolume work Conics by the Greek mathematician Apollonius, who lived from c. 262 - 190 BCE.

One theory is that the Ancient Greeks began studying these shapes - ellipses, parabolas, hyperbolas - as they were using sundials to study the sun's apparent movement.
• cant the foci points be on the minor radius as well? Seems like it would work exactly the same
• In an ellipse, foci points have a special significance. Any ray emitted from one focus will always reach the other focus after bouncing off the edge of the ellipse (This is why whispering galleries are in the shape of an ellipsoid). The foci can only do this if they are located on the major axis.
• How do I get the directrix?
• Go to the next section in the lessons where it covers directrix.
• The equations of circle, ellipse, parabola or hyperbola are just equations and not function right? Is it because when y is squared, the function cannot be defined?
• Almost correct. The equation of a parabola could be a function (a quadratic function in 𝑥). However, the rest cannot be functions.
• What is the eccentricity of an ellipse?
(1 vote)
• Eccentricity is a measure of how close the ellipse is to being a perfect circle. A perfect circle has eccentricity 0, and the eccentricity approaches 1 as the ellipse stretches out, with a parabola having eccentricity exactly 1.

You can compute the eccentricity as c/a, where c is the distance from the center to a focus, and a is the length of the semimajor axis.
• Is the eccentricity of an ellipse the ratio between a point's distance from the nearest focus and its distance from the nearest directrix? Does this generalize to other conic sections?
• Yes, and yes. Every conic section can be defined by a point (taken as the focus), a line (taken as the directrix), and a positive number for the eccentricity. The conic section is the set of all points whose distance from the focus is in constant ratio with the distance from the directrix. That constant ratio is the eccentricity.

The eccentricity of a circle is 0 (though we must either take the directrix to be infinitely far away, or take the circle's radius to be 0).

The eccentricity of any other ellipse is between 0 and 1, where higher eccentricity gives more stretched-out ellipses.

The eccentricity of a parabola is 1.

The eccentricity of a hyperbola is greater than 1.