If PQ is a double ordinate of the hyperbola $\left({\mathrm{x}}^2/{\mathrm{a}}^2\right)-\left({\mathrm{y}}^2/{\mathrm{b}}^2\right)=1$ such that OPQ is an equilateral triangle, O being the centre of the hyperbola, then the least value of eccentricity is

$$\begin{gathered} Let the hyperbola be \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \\ Let P=(asec\theta ,b\tan \theta ),Q=(asec\theta ,-b\tan \theta ) be end points of double ordinate \\ Since ∆OPQ is an equilateral triangle then \end{gathered}$$

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