﻿ Focus of Ellipse. The formula for the focus and ...

# Focus of Ellipse

Formula and examples for Focus of Ellipse

#### What is a focus of an ellipse?

An ellipse has 2 foci (plural of focus). In the demonstration below, these foci are represented by blue tacks . These 2 foci are fixed and never move.

Now, the ellipse itself is a new set of points. To draw this set of points and to make our ellipse, the following statement must be true: if you take any point on the ellipse, the sum of the distances to those 2 fixed points ( blue tacks ) is constant. We explain this fully here.

Full lesson on what makes a shape an ellipse here .

### Formula for the focus of an Ellipse

Diagram 1

The formula generally associated with the focus of an ellipse is $$c^2 = a^2 - b^2$$ where $$c$$ is the distance from the focus to center, $$a$$ is the distance from the center to a vetex and $$b$$ is the distance from the center to a co-vetex .

### Example of Focus

In diagram 2 below, the foci are located 4 units from the center. All that we need to know is the values of $$a$$ and $$b$$ and we can use the formula $$c^2 = a^2- b^2$$ to find that the foci are located at $$(-4,0)$$ and $$(4,0)$$ .

Diagram 2

### Practice Problems

##### Problem 1

Use the formula for the focus to determine the coordinates of the foci.

Use the formula and substitute the values:

$c^2 = a^2 - b^2 \\ c^2 = 5^2 - 3^2 \\ c^2 = 25 - 9 = 16 \\ c = \sqrt{16} \\ c = \boxed{4} \\ \text{ foci : } (0,4) \text{ & }(0,-4)$

##### Problem 2

Use the formula for the focus to determine the coordinates of the foci.

Use the formula and substitute the values:

$c^2 = a^2 - b^2 \\ c^2 = 10^2 - 6^2 \\ c^2 = 100 - 36 = 64 \\ c = \sqrt{64} \\ c = \boxed{8} \\ \text{ foci : } (0,8) \text{ & }(0,-8)$

##### Problem 3

Use the formula for the focus to determine the coordinates of the foci.

Use the formula and substitute the values:

$c^2 = a^2 - b^2 \\ c^2 = 25^2 - 7^2 \\ c^2 = 625 - 49 \\ c^2 = 576 \\ c = \sqrt{576} \\ c = \boxed{44} \\ \text{ foci : } (0,24) \text{ & }(0,-24)$

### Focus of Ellipse from the Equation

The problems below provide practice finding the focus of an ellipse from the ellipse's equation.
All practice problems on this page have the ellipse centered at the origin.

Click here for practice problems involving an ellipse not centered at the origin.
##### Problem 4

Use the formula for the focus to determine the coordinates of the foci.

$25x^2 + 9y^2 = 225$

First, rewrite the equation in stanadard form, then use the formula and substitute the values. :

##### Problem 5

Use the formula for the focus to determine the coordinates of the foci.

$100x^2 + 36y^2 = 3,600$

First, rewrite the equation in stanadard form, then use the formula and substitute the values. :