Question

The eccentricity of the hyperbola whose latus-return is $8$ and length of the conjugate axis is equal to half the distance between the foci, is

• A. $\frac{4}{3}$
• B. $\frac{4}{\surd 3}$
• C. $\frac{2}{\surd 3}$
• D. $Noneofthese$

Answer 1

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

Given that the length of the latus rectum is $8$ and length of the conjugate axis is equal to half the distance between the foci.

$\Rightarrow \frac{2b^2}{a}=8$ and $2b=\frac{1}{2}\left(2ae\right)$

$\therefore \frac{2}{a}{\left(\frac{ae}{2}\right)}^2=8$

$\Rightarrow a{e}^2=16$ ...(1)

We have $\frac{2b^2}{a}=8$

$\Rightarrow b^2=4a$

$\Rightarrow a^2(e^2-1)=4a$

$\Rightarrow a{e}^2-a=4$

$\Rightarrow 16-a=4$ (by (1))

$\Rightarrow a=12$

Substitute $a=12$ in (1)

$\Rightarrow 12e^2=16$

$\Rightarrow e^2=\frac{4}{3}$

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