# Hyperbola: Formula

How to write the equation of a hyperbola by using its foci and transverse axis: formulas, examples, and their solutions.

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

If the vertices of the hyperbola are (±*a*, 0),

and if the foci are (±*c*, 0),

then the equation of the hyperbola is*x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1.

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

## Example 1: Foci (3, 0) and (-3, 0), Transverse Axis 4, Hyperbola?

The foci are (3, 0) and (-3, 0).

Then *c* = 3.

The transverse axis is 4.

And the *y* values of the foci are the same.

So the transverse axis is 2*a*.

So 2*a* = 4.

Hyperbola - Transverse axis

Divide both sides by 2.

Then *a* = 2.

*a* = 2*c* = 3

So 2^{2} + *b*^{2} = 3^{2}.

2^{2} = 4

3^{2} = 9

Move 4 to the right side.

Then *b*^{2} = 5.

Instead of finding the value of *b*,

directly use *b*^{2} = 5

to write the equation of the hyperbola.

*a* = 2*b*^{2} = 5

So the equation of the hyperbola is*x*^{2}/2^{2} - *y*^{2}/5 = 1.

2^{2} = 4

So [*x*^{2}/4 - *y*^{2}/5 = 1] is the answer.

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

Its foci are (3, 0) and (-3, 0).

And its transverse axis is 2⋅2 = 4.

## Example 2: Foci (-3, 2) and (5, 2), Transverse Axis 6, Hyperbola?

Lightly draw the given conditions.

It says the foci are (-3, 2) and (5, 2).

And the transverse axis is 6.

So draw a hyperbola like this.

The foci are (-3, 2) and (5, 2).

And the distance between the foci is 2*c*.

So 2*c* = 5 - (-3).

-(-3) = +3

5 + 3 = 8

Divide both sides by 2.

Then *c* = 4.

*c* = 4

And the *y* values of the foci are the same.

So the original foci are (-4, 0) and (4, 0).

But the given foci are (-3, 2) and (5, 2).

So the foci are under a translation.

Use (4, 0) and (5, 2) to find the translation:

(5, 2) = (4 + 1, 0 + 2).

So the translation is

(*x*, *y*) → (*x* + 1, *y* + 2).

Translation of a point

Next, it says the transverse axis is 6.

And the *y* values of the foci are the same.

So the transverse axis is 2*a*.

So 2*a* = 6.

Divide both sides by 2.

Then *a* = 3.

*a* = 3*c* = 4

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

3^{2} = 9

4^{2} = 16

Move 9 to the right side.

Then *b*^{2} = 7.

Instead of finding the value of *b*,

directly use *b*^{2} = 7

to write the equation of the hyperbola.

*a* = 3*b*^{2} = 7

The hyperbola is under the translation

(*x*, *y*) → (*x* + 1, *y* + 2).

So the equation of the hyperbola is

(*x* - 1)^{2}/3^{2} - (*y* - 2)^{2}/7 = 1.

Translation of a function

3^{2} = 9

So (*x* - 1)^{2}/9 - (*y* - 2)^{2}/7 = 1.

This is the graph of (*x* - 1)^{2}/9 - (*y* - 2)^{2}/7 = 1.

Its foci are (-4 + 1, 0 + 2) = (-3, 2)

and (4 + 1, 0 + 2) = (5, 2).

And its transverse axis is 2⋅3 = 6.

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

If the vertices of the hyperbola are (0, ±*b*),

and if the foci are (0, ±*c*),

then the equation of the hyperbola is*y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1.

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

## Example 3: Foci (0, 2) and (0, -2), Transverse Axis 2, Hyperbola?

The foci are (0, 2) and (0, -2).

Then *c* = 2.

The transverse axis is 2.

And the *x* values of the foci are the same.

So the transverse axis is 2*b*.

So 2*b* = 2.

Hyperbola - Transverse axis

Divide both sides by 2.

Then *b* = 1.

*b* = 1*c* = 2

So *a*^{2} + 1^{2} = 2^{2}.

1^{2} = 1

2^{2} = 4

Move +1 to the right side.

Then *a*^{2} = 3.

Instead of finding the value of *a*,

directly use *a*^{2} = 3

to write the equation of the hyperbola.

*a*^{2} = 3*b* = 1

So the equation of the hyperbola is*y*^{2}/1^{2} - *x*^{2}/3 = 1.

1^{2} = 1

So [*y*^{2} - *x*^{2}/3 = 1] is the answer.

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

Its foci are (0, 2) and (0, -2).

And its transverse axis is 2⋅1 = 2.