How do I determine the asymptotes of a hyperbola?

1 Answer
Sep 28, 2014

If a hyperbola has an equation of the form $\frac{{x}^{2}}{{a}^{2}} - \frac{{y}^{2}}{{b}^{2}} = 1$ $\left(a > 0 , b > 0\right)$, then its slant asymptotes are $y = \pm \frac{b}{a} x$.

Let us look at some details.

By observing,

$\frac{{x}^{2}}{{a}^{2}} - \frac{{y}^{2}}{{b}^{2}} = 1$

by subtracting $\frac{{x}^{2}}{{a}^{2}}$,

$R i g h t a r r o w - \frac{{y}^{2}}{{b}^{2}} = - \frac{{x}^{2}}{{a}^{2}} + 1$

by multiplying by $- {b}^{2}$,

$R i g h t a r r o w {y}^{2} = \frac{{b}^{2}}{{a}^{2}} {x}^{2} - {b}^{2}$

by taking the square-root,

$R i g h t a r r o w y = \pm \sqrt{\frac{{b}^{2}}{{a}^{2}} {x}^{2} - {b}^{2}} \approx \pm \sqrt{\frac{{b}^{2}}{{a}^{2}} {x}^{2}} = \pm \frac{b}{a} x$

(Note that when $x$ is large, $- {b}^{2}$ is negligible.)

Hence, its slant asymptotes are $y = \pm \frac{b}{a} x$.
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