Asymptotes of Hyperbolas - Practice Questions & MCQ

Asymptotes of Hyperbola:

Asymptote of a curve is a straight line such that the distance between the curve and the line approaches to zero when one or both of $x$ - and $y$-coordinate approach infinity.

For example,Asymptote of the curve $y=1 / x$ is straight line $y=0$ and $x=0$.

An asymptote of any hyperbola is a straight line that touches it at infinity.
The equation of the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are $y= \pm \frac{b}{a} x$ or $\frac{x}{a} \pm \frac{y}{b}=0$

To find the asymptotes of the hyperbola,

Let the straight line $y=m x+c$ is asymptotes to the hyperbola

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

put the value of $y$ in the Eq. of hyperbola

$
\begin{aligned}
& \frac{x^2}{a^2}-\frac{(m x+c)^2}{b^2}=1 \\
& \left(\mathrm{a}^2 m^2-b^2\right) x^2+2 a^2 m c x+a^2\left(c^2+b^2\right)=0
\end{aligned}
$

since, asymptotes touch the hyperbola at infinity, so both roots of the quadratic equation must be infinite and condition for which is coefficient of $x^2$ and $x$ must be zero.

$
\begin{array}{lc}
\therefore & a^2 \mathrm{~m}^2-\mathrm{b}^2=0 \\
\text { and } & 2 \mathrm{a}^2 \mathrm{mc}=0
\end{array}
$
$
\mathrm{m}= \pm \frac{\mathrm{b}}{\mathrm{a}} \text { and } \mathrm{c}=0
$

put the value of $m$ in $y=m x+c$

$
\mathrm{y}= \pm \frac{\mathrm{b}}{\mathrm{a}} \mathrm{x} \quad \text { or } \quad \frac{\mathrm{x}}{\mathrm{a}} \pm \frac{\mathrm{y}}{\mathrm{~b}}=0
$
TIP:
The angle between the asymptotes of the hyperbola $\frac{y^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \tan ^{-1}\left(\frac{b}{a}\right)$
If the angle between the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \theta$ then $e=\sec \theta$