Latus rectum of a hyperbola is 8 and its conjugate axis is equal to half the distance between the foci. What is the eccentricity of the hyperbola.

The correct option is C

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

Given that,length of latus rectum=8

i.e,$\frac{2b^2}{a}=8$

i.e.,$b^2=4a$ ...(1)

Standard equation of a hyperbola is,

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

For this hyperbola,

Latus rectum ,$L=\frac{2b^2}{a}$

Distnce between the foci = 2ae.

Transverse axis length =2a.

Conjugate axis length =2b.

Given that

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

i.e.,$2b=ae$ .....(2)

Also in a hyperbola

$b^2=a^2(e^2-1)$ .....(3)

$4a=a^2(e^2-1)$ (using (1))

$a=\frac{4}{e^2-1}$

$b=2\sqrt{a}=\frac{2.2}{\sqrt{e^2-1}}=\frac{4}{\sqrt{e^2-1}}$

Using(2)

2.$\frac{4}{\sqrt{e^2-1}}=\frac{4}{e^2-1}.e$

$2\sqrt{(e^2-1)}=e$

$4(e^2-1)=e^2$

$4e^2-4=e^2$

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

Hence ans (c)