Question

$PQ$ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola, then the range of the eccentricity $e$ of the hyperbola is

• A. $1<e<2$
• B. $1<e<\sqrt{2}$
• C. $e\ge \frac{2}{\sqrt{3}}$
• D. $e<4$

Answer 1

The correct option is C $e\ge \frac{2}{\sqrt{3}}$


Let double ordinate $PQ$ be such that $P=(a\sec \theta ,b\tan \theta )$ and $Q=(a\sec \theta ,-b\tan \theta )$ and $O$ is the centre $(0,0)$.
From $△OPR$
$\tan {30}^{\circ }=\frac{b\tan \theta }{a\sec \theta }$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{b}{a}\sin \theta$
$\Rightarrow \frac{3b^2}{a^2}={\text{cosec}}^2\theta$
$\Rightarrow 3(e^2-1)={\text{cosec}}^2\theta \ge 1$
$\Rightarrow e\ge \frac{2}{\sqrt{3}}$