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. $\sqrt{3}$

Answer 1

The correct option is C $\frac{2}{\sqrt{3}}$

Length of latus rectum $=\frac{2b^2}{a}=8$

And $2b=\frac{1}{2}2ae$

Or $b=\frac{ae}{2}.......\left(1\right)$

The eccentricity of the hyperbola is given by:

$e=\sqrt{1+\frac{b^2}{a^2}}$

Squaring both sides we get:

$e^2=1+\frac{e^2}{4}$ ....... Using $\left(1\right)$

Or $\frac{3e^2}{4}=1$

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

Hence, option C is correct.