Question

How do I determine the asymptotes of a hyperbola?

Answer 1

If a hyperbola has an equation of the form `{x^2}/{a^2}-{y^2}/{b^2}=1` `(a>0, b>0)`, then its slant asymptotes are `y=pm b/ax`.

Let us look at some details.

By observing,

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

by subtracting `{x^2}/{a^2}`,

`Rightarrow -{y^2}/{b^2}=-{x^2}/{a^2}+1`

by multiplying by `-b^2`,

`Rightarrow y^2={b^2}/{a^2}x^2-b^2`

by taking the square-root,

`Rightarrow y=pm sqrt{ {b^2}/{a^2}x^2-b^2 } approx pm sqrt{{b^2}/{a^2}x^2}=pm b/a x`

(Note that when `x` is large, `-b^2` is negligible.)

Hence, its slant asymptotes are `y=pm b/a x`.