Question

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

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

Answer 1

The Correct Option is D

$\because \Delta OPQ$ is equilateral, $OP = PQ$ $\Rightarrow a^{2}\,sec^{2}\,\theta+b^{2}\,tan^{2}\,\theta=\left(2b\,tan\,\theta\right)^{2}$ $\Rightarrow a^{2}\,sec^{2}\,\theta+3b^{2}\,tan^{2}\,\theta$ $\Rightarrow sin^{2}\,\theta= \frac{a^{2}}{3b^{2}}$ Now, $sin^{2}\,\theta <1$ $\Rightarrow \frac{a^{2}}{3b^{2}} >1$ $\Rightarrow \frac{b^{2}}{a^{2}}>\frac{1}{3} \Rightarrow 1+\frac{b^{2}}{a^{2}} > \frac{4}{3} \Rightarrow e^{2}>\frac{4}{3} \Rightarrow e> \frac{2}{\sqrt{3}}$