Question

$P Q$ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $\triangle O P Q$ is an equilateral triangle, $O$ being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies

• A. $1 < e< 2 / \sqrt{3}$
• B. $e=2 / \sqrt{3}$
• C. $e=2 \sqrt{3}$
• D. $e >2 / \sqrt{3}$

Answer 1

Answer: (D)

$\tan 30^{\circ}=\frac{b \tan \theta}{a \sec \theta}$
$ \Rightarrow \frac{b}{a}=\frac{1}{\sin \theta \sqrt{3}} $
$ e=\sqrt{1+\frac{1}{3 \sin ^2 \theta}}>\sqrt{1+\frac{1}{3}}$

$\Rightarrow e>\frac{2}{\sqrt{3}}\left(0<\sin ^2 \theta<1\right)$