# Hyperbola: Asymptote

How to find the asymptote of a hyperbola: formula, 2 examples, and their solutions.

## FormulaHorizontal Hyperbola

The asymptote is a line

the graph follows.

So, for a hyperbola,

there are two asymptotes.

Hyperbola: Equation

For the horizontal hyperbola

x^{2}/a^{2} - y^{2}/b^{2} = 1,

the asymptotes are

y = ±[b/a]x.

## Examplex^{2}/9 - y^{2}/16 = 1

Solution

Solution (Detail)

9 = 3^{2}

16 = 4^{2}

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

a = 3

b = 4

Then the asymptotes are

y = ±[4/3]x.

So

y = ±[4/3]x

is the answer.

## FormulaVertical Hyperbola

For the vertical hyperbola

y^{2}/a^{2} - x^{2}/b^{2} = 1,

the asymptotes are

y = ±[a/b]x.

a and b are switched.

## Exampley^{2} - 9x^{2} = 9

Solution

Solution (Detail)

To make the right side 1,

divide both sides by 9.

9 = 3^{2}

-x^{2}

= -x^{2}/1

= -x^{2}/1^{2}

y^{2}/3^{2} - x^{2}/1^{2} = 1

a = 3

b = 1

Then the asymptotes are

y = ±[3/1]x.

√x

3/1 = 3

So

y = ±3x

is the answer.