# Ellipse: Foci

How to find the foci of an ellipse: formulas, examples, and their solutions.

## Formula: If *a* > *b*

For the ellipse *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1,

if *a* > *b*,

then the foci are (±*c*, 0)

and *c*^{2} = *a*^{2} - *b*^{2}.

## Example 1: Foci of *x*^{2}/25 + *y*^{2}/16 = 1

Change the equation in standard form.

25 = 5^{2}

16 = 4^{2}

So *x*^{2}/5^{2} + *y*^{2}/4^{2} = 1.

*a* = 5*b* = 4

And *a* > *b*.

So *c*^{2} = 5^{2} - 4^{2}.

5^{2} = 25

4^{2} = 16

25 - 16 = 9

So *c*^{2} = 9.

Square root both sides.

Then *c* = ±3.

*c* = ±3

And *a* > *b*.

So the foci are (±3, 0).

This is the graph of *x*^{2}/5^{2} + *y*^{2}/4^{2} = 1.*c* = ±3

And *a* > *b*. (5 > 4)

So the foci are (±3, 0).

## Formula: If *a* < *b*

For the ellipse *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1,

if *a* < *b*,

then the foci are (0, ±*c*)

and *c*^{2} = *b*^{2} - *a*^{2}.

## Example 2: Foci of 9*x*^{2} + 4*y*^{2} = 36

Change the equation in standard form.

Divide both sides by 36.

9/36 = 1/4

4/36 = 1/9

4 = 2^{2}

9 = 3^{2}

So *x*^{2}/2^{2} + *y*^{2}/3^{2} = 1.

*a* = 2*b* = 3

And *a* < *b*.

So *c*^{2} = 3^{2} - 2^{2}.

3^{2} = 9

2^{2} = 4

9 - 4 = 5

So *c*^{2} = 5.

Square root both sides.

Then *c* = ±√5.

*c* = ±√5

And *a* < *b*.

So the foci are (0, ±√5).

This is the graph of *x*^{2}/2^{2} + *y*^{2}/3^{2} = 1.*c* = ±√5

And *a* < *b*. (2 < 3)

So the foci are (0, ±√5).