Question

The distance of the point $(\sqrt{6}sec\theta ,\sqrt{3}tan\theta )$ on the hyperbola from the centre of hyperbola is 3 if $\theta$ is :

• A. $\frac{\pi }{12}$
• B. $\frac{\pi }{8}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (C)

$\frac{\pi }{6}$

$P\left(\sqrt{6}\sec \theta ,\sqrt{3}\tan \theta \right)$

0 is centre of Hyperbola is (0,0)

$OP=\sqrt{6{\sec }^2\theta +3{\tan }^2\theta }=3$

Square both side, we get

$\begin{array}{c}\Rightarrow 6{\sec }^2\theta +3\left({\sec }^2\theta -1\right)=3\\ \Rightarrow 6{\sec }^2\theta +3{\sec }^2\theta -3=9\Rightarrow 9{\sec }^2\theta =12\Rightarrow 3{\sec }^2\theta =4\\ \sqrt{{\sec }^2\theta }=\sqrt{\frac{4}{3}}\\ \sec \theta =\frac{2}{\sqrt{3}}\\ \cos \theta =\frac{\sqrt{3}}{2}\Rightarrow \theta =\frac{\pi }{6}\end{array}$