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

The correct option is C 23

Given that,length of latus rectum $=8$

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

$\Rightarrow b^2=4a$ ... (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 $=2ae$

Length of transverse axis $=2a$

Length of conjugate axis $=2b$

Given that

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

$\Rightarrow 2b=ae$ ... (2)

Also in a hyperbola,

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

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

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

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

Using (2)

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

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

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

$\Rightarrow 4e^2-4=e^2$

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