Question

The angle between the asymptotes of the hyperbola $x^{2}-3 y^{2}=3$ is

• A. $\frac{\pi}{6}$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{2}$

Answer 1

Answer: (C)

Given equation of hyperbola is
$ x^{2}-3 y^{2}=3 $
$\Rightarrow \frac{x^{2}}{3}-\frac{y^{2}}{1}=1$
Here, $a^{2}=3$ and $b^{2}=1, a>b$
Now, the equation of asymptote of this hyperbola is,
$y=\pm \frac{b}{a} x$
$\Rightarrow y=\pm \frac{1}{\sqrt{3}} x$
$\Rightarrow y=\frac{x}{\sqrt{3}}\,\,\,...(i)$
and $y=\frac{-x}{\sqrt{3}}\,\,\,...(ii)$
Let slope of asymptote (i) is, $m_{1}=\frac{1}{\sqrt{3}}$
and slope of asymptote (ii) is, $m_{2}=\frac{-1}{\sqrt{3}}$
Let $\theta$ be the angle between both asymptote, then
$\tan \theta=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|=\left|\frac{1 / \sqrt{3}+1 / \sqrt{3}}{1-1 / 3}\right| $
$\Rightarrow \tan \theta=\left|\frac{2 / \sqrt{3}}{2 / 3}\right|=\sqrt{3} $
$\Rightarrow \tan \theta=\tan 60^{\circ} $
$\Rightarrow \theta=\pi / 3$