# Hyperbola: Foci

How to find the foci of a hyperbola: formulas, examples, and their solutions.

## Formula: Foci of *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1

For the parabola *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1,

the foci are (±*c*, 0)

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

[*c*^{2} = *a*^{2} + *b*^{2}] came from

the proof of the hyperbola formula.

Hyperbola - Proof of the formula *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1

## Example 1: Foci of *x*^{2}/9 - *y*^{2}/16 = 1

Change the equation in standard form.

9 = 3^{2}

16 = 4^{2}

So *x*^{2}/3^{2} - *y*^{2}/4^{2} = 1.

*a* = 3*b* = 4

So *c*^{2} = 3^{2} + 4^{2}.

3^{2} = 9

4^{2} = 16

9 + 16 = 25

So *c*^{2} = 25.

Square root both sides.

Then *c* = ±5.

*c* = ±5

And the *x*^{2} term is (+).

So the foci are (±5, 0).

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

And the *x*^{2} term is (+).

So the foci are (±5, 0).

## Formula: Foci of *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1

For the parabola *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1,

the foci are (0, ±*c*)

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

[*c*^{2} = *a*^{2} + *b*^{2}] came from

the proof of the hyperbola formula.

Hyperbola - Proof of the formula *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1

## Example 2: Foci of *x*^{2} - *y*^{2} = -4

Change the equation in standard form.

Divide both sides by -4.

Switch the order of the terms in the left side.

4 = 2^{2}

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

*a* = 2*b* = 2

So *c*^{2} = 2^{2} + 2^{2}.

2^{2} = 4

4 + 4 = 8

So *c*^{2} = 8.

Square root both sides.

Then *c* = ±√8.

√8

= √2^{2}⋅2

= 2√2

Simplify a radical

So *c* = ±2√2.

*c* = ±2√2

And the *y*^{2} term of [*y*^{2}/2^{2} - *x*^{2}/2^{2} = 1] is (+).

So the foci are (0, ±2√2).

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

And the *y*^{2} term is (+).

So the foci are (0, ±2√2).