The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is___ .

\frac{4}{3}

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\frac{4}{\sqrt{3}}

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\frac{2}{\sqrt{3}}

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\sqrt{3}

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Solution

The correct option is C $\frac{2}{\sqrt{3}}$
Let the hyperbola be $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
We have , $\frac{2 b^{2}}{a} = 8$ and $2 b = a e$
$\Rightarrow b^{2} = 4 a a n d 2 b = a e$
Consider, $2 b = a e \Rightarrow 4 b^{2} = a^{2} e^{2} \Rightarrow 4 a^{2} (e^{2} - 1) = a^{2} e^{2} \Rightarrow 4 e^{2} - 4 = e^{2} \Rightarrow 3 e^{2} = 4 e = \frac{2}{\sqrt{3}}$

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