Let LL’ be the latus rectum through the focus of the hyperbola x2a2−y2b2=1 and A’ be the farther vertex. If △A’LL’ is equilateral, then the eccentricity of the hyperbola is (axex are coordinate axes)

Let LL' be the latus rectum through the focus of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and A' be the farther vertex. If $△ A' LL'$ is equilateral, then the eccentricity of the hyperbola is (axex are coordinate axes)

A
$\sqrt{3}$
B
$\sqrt{3} + 1$
C
$\frac{\sqrt{3} + 1}{\sqrt{2}}$
D
$\frac{\sqrt{3} + 1}{\sqrt{3}}$

Solution:

$\tan 30^{\circ} = \frac{b^{2}}{a^{2} (e + 1)} \Rightarrow e = \frac{\sqrt{3} + 1}{\sqrt{3}} w h e r e b^{2} = a^{2} (e^{2} - 1)$

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