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

$Given hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Let double odinate PQ be such that $\mathrm{P}=(\mathrm{asec}\mathrm{\theta },\mathrm{btan}\mathrm{\theta })$, and $\mathrm{Q}(\mathrm{asec}\mathrm{\theta },-\mathrm{btan}\mathrm{\theta })$ and O is centre (0, 0).

From $\mathrm{△}\mathrm{OPR}$

$\begin{array}{l}\Rightarrow \tan {30}^0=\frac{\mathrm{btan}\mathrm{\theta }}{\mathrm{asec}\mathrm{\theta }}\\ \Rightarrow 3\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}={\mathrm{cosec}}^2\mathrm{\theta }\\ \Rightarrow 3\left({\mathrm{e}}^2-1\right)={\mathrm{cosec}}^2\mathrm{\theta }\ge 1\\ \Rightarrow {\mathrm{e}}^2-1\ge \frac{1}{3}\\ \Rightarrow e^2\ge \frac{4}{3}\\ \Rightarrow \mathrm{e}\ge \frac{2}{\sqrt{3}}\end{array}$