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)

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

Answer 2

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 $2b=\frac{1}{2}\left(2ae\right)$ $\Rightarrow 2b = ae$ $\Rightarrow 4b^{2} = a^{2}e^{2}$ $\Rightarrow\, 4\left(\frac{b^{2}}{a^{2}}\right)=e^{2}$ $\Rightarrow 4\left(e^{2}-1\right)=e^{2} \left[\because\, b^{2}=a^{2} \left(e^{2}-1\right)\right]$ $\Rightarrow 3e^{2} = 4$ $\Rightarrow e=\frac{2}{\sqrt{3}}$