# Hyperbola: Formula, Foci, Transverse Axis

How to solve the hyperbola problems: definition, formulas (equations, foci, transverse axis), proof, examples, and their solutions.

## Definition

An hyperbola is the set of points

whose difference of the distances from the foci

is constant.

(= transverse axis, shown below)

The difference of the blue segments' lengths is constant.

## Formula 1

The transverse axis is the segment

whose endpoints are the vertices of the hyperbola.

(blue, 2*a*)

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}.

If the *y*^{2} term is (-),

(and if the right side is 1, not -1,)

the foci are on the *x*-axis.

So, if the *y*^{2} term is (-), use this formula.

The shape of a hyperbola is different

from the shape of a parabola.

## Proof: Formula 1

Set [distance between (*x*, *y*) and (-*c*, 0)]

- [distance between (*x*, *y*) and (*c*, 0)]

= ±2*a*.

Move the second radical term to the right side.

Square both sides.

Square of a sum

Cancel the gray terms.

Then arrange the terms

so that only the remaining radical term

is on the left side.

Divide both sides by 4.

Square both sides.

Square of a sum

Cancel the gray terms.

Make (*a*^{2} - *c*^{2}) by factoring.

Then put -*b*^{2}

into *a*^{2} - *c*^{2}.

Divide both sides by *a*^{2}(-*b*^{2}).

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

And the relationship between *a*, *b*, and *c* is*a*^{2} + *b*^{2} = *c*^{2}.

## Example 1

To find *a*, *b*, and *c*,

change the equation into the standard form.*x*^{2}/3^{2} - *y*^{2}/4^{2} = 1

Then *a* = 3 and *b* = 4.

*y*^{2} term is (-).

So (transverse axis) = 2⋅3 = 6.

Use *a*^{2} + *b*^{2} = *c*^{2}

to find *c*.*c*^{2} = *a*^{2} + *b*^{2}

= 3^{2} + 4^{2}*c* = ±5

So the foci are (±5, 0).

This is the graph of the hyperbola

you've solved above.

## Example 2

Roughly draw the given conditions.

Foci: (-1, -1), (5, -1)

The distance between the foci is 2|*c*|.

The foci are located horizontally.

So the transverse axis is 2*a* = 2.

The distance between the foci (-1, -1), (5, -1)

is 2|*c*|.

So 2|*c*| = 5 - (-1) = 6.*c* = ±3

The foci are not (±3, 0).

So the hyperbola is under a translation.

Translation of a point

So use *c* = 3

to find the changes of *x* and *y*:

(5, -1) = (3 + 2, -1).

So the hyperbola is under the translation

(*x*, *y*) → (*x* + 2, *y* - 1).

The transverse axis is 2*a* = 2.

So *a* = 1.

Use *a*^{2} + *b*^{2} = *c*^{2}

to find *b*.

1^{2} + *b*^{2} = 3^{2}*b*^{2} = 8

Write the equation of the hyperbola

using the above conditions.

(*x* - 2)^{2}/1^{2} - (*y* + 1)^{2}/8 = 1

Translation of a function

See how (*x* - 2)^{2}/1^{2} - (*y* + 1)^{2}/8 = 1 looks like.

The graph is determined by

its foci (+3 + 2, -1), (-3 + 2, -1)

and its transverse axis 2⋅1.

## Formula 2

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

then the transverse axis is vertical.

(green, 2*b*)

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}.

If the *x*^{2} term is (-), the foci are on the *y*-axis.

So, if the *x*^{2} term is (-), use this formula.

## Proof: Formula 2

Set [distance between (*x*, *y*) and (0, -*c*)]

- [distance between (*x*, *y*) and (0, *c*)]

= ±2*b*.

Move the second radical term to the right side.

Square both sides.

Square of a sum

Cancel the gray terms.

Then arrange the terms

so that only the remaining radical term

is on the left side.

Divide both sides by 4.

Square both sides.

Square of a sum

Cancel the gray terms.

Make (*b*^{2} - *c*^{2}) by factoring.

Then put -*a*^{2}

into *b*^{2} - *c*^{2}.

Switch the left side terms.

And divide both sides by *b*^{2}(-*a*^{2}).

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

And the relationship between *a*, *b*, and *c* is*a*^{2} + *b*^{2} = *c*^{2}.

## Example 3

To find *a*, *b*, and *c*,

change the equation into the standard form.*y*^{2}/2^{2} - *x*^{2}/2^{2} = 1

Then *a* = *b* = 2.

*x*^{2} term is (-).

So (transverse axis) = 2⋅2 = 4.

(2 from *b*)

Use *a*^{2} + *b*^{2} = *c*^{2}

to find *c*.*c*^{2} = *a*^{2} + *b*^{2}

= 2^{2} + 2^{2}*c* = ±2√2

So the foci are (0, ±2√2).

This is the graph of the hyperbola

you've solved above.