Question

The eccentricity of the hyperbola whose latus rectum is $8$ and conjugate axis is equal to half of the distance between the foci is

• A. $\frac{4}{3}$
• B. $\frac{4}{\sqrt{3}}$
• C. $\frac{2}{\sqrt{3}}$
• D. None of these

Answer 1

Answer: (C)

Let equation of hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Given, $\frac{2b^2}{a}=8$
$\Rightarrow \frac{b^2}{a}=4$
and $2b=\frac{1}{2}\left(2ae\right)$
$\Rightarrow 2b=ae$
$\Rightarrow 4b^2=a^2{e}^2$
$\Rightarrow 4\left(\frac{b^2}{a^2}\right)=e^2$
$\Rightarrow 4\left(e^2-1\right)=e^2\left[\because b^2=a^2\left(e^2-1\right)\right]$
$\Rightarrow 3e^2=4$
$\Rightarrow e=\frac{2}{\sqrt{3}}$