If the latus rectum of a hyperbola through one focus subtends an angle $60^{\circ}$ at the other focus, then its eccentricity is

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Aicharya · Accepted Answer

Given, latus rectum of one hyperbola subtends an angle of $60^{\circ}$ at other follows,

From $\triangle A B C$, $\begin{array}{llll} & \tan 30^{\circ} =\frac{b^2 / a}{2 a e} \\ \Rightarrow & \frac{1}{\sqrt{3}} =\frac{b^2}{2 a^2 e} \\ & \text {Also } b^2 =a^2\left(e^2-1\right) \\ & \text {So, } \frac{1}{\sqrt{3}} =\frac{a^2\left(e^2-1\right)}{2 a^2 e} \\ \Rightarrow & 2 e =\sqrt{3}\left(e^2-1\right) \\ \Rightarrow & e =\sqrt{3} \end{array}$

If the latus rectum of a hyperbola through one focus subtends an angle \(60^{\circ}\) at the other focus,…

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