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

$\begin{array}{c}\text{let P}\left(a\sec \theta ,b\tan \theta \right)\text{\hspace{0.17em}and Q}\left(asec\theta ,-btan\theta \right)\text{ be end points of the double ordinate}\\ \text{and C}\left(0,0\right)\text{\hspace{0.17em}is the center of the hyperbola}\\ PQ=2b\tan \theta \\ OQ=OP=\sqrt{a^2{\sec }^2\theta +b^2{\tan }^2\theta }\\ \text{Since OPQ is an equaliateral triangle}\\ OP=OQ=PQ\\ 4b^2{\tan }^2\theta =a^2{\sec }^2\theta +b^2{\tan }^2\theta \\ 3b^2{\tan }^2\theta =a^2\left(1+{\tan }^2\theta \right)\\ 3a^2\left(e^2-1\right){\sin }^2\theta =a^2\\ \frac{1}{e^2-1}<3\\ e>\frac{2}{\sqrt{3}}\end{array}$