Question

If the latus rectum of a hyperbola forms an equilateral triangle with the vertex at the centre of the hyperbola, then its eccentricity e =

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$Given,\angle SOL={30}^{\circ }$

$\therefore tan{30}^{\circ }=\frac{SL}{OS}\Rightarrow \frac{1}{\sqrt{3}}=\frac{b^2}{a.ae}\Rightarrow a^{2}e=\sqrt{3}{a}^2(e^2-1)\Rightarrow \sqrt{3}{e}^2-e-\sqrt{3}=0\Rightarrow e=\frac{\sqrt{13}+1}{2\sqrt{3}}$