The eccentricity of the hyperbola whose latus rectum is $8$ and conjugate axis is equal to half of the distance between the foci is

The correct option is D $\frac{2}{\sqrt{3}}$
Given: $2b=\frac{1}{2}\left(2ae\right)$
$\left[{}_{\text{and length of conjugate axis}=2b}^{\because \text{Distance between foci}=2ae}\right]$
$\Rightarrow 2b=ae$
Squaring on both sides
$\Rightarrow 4b^2=a^2{e}^2$
Divide by $a^2$ on both sides
$\Rightarrow 4\left(\frac{b^2}{a^2}\right)=e^2$
$\Rightarrow 4(e^2-1)=e^2[\because b^2=a^2(e^2-1\left)\right]$
$\Rightarrow 4e^2-4=e^2$
$\Rightarrow 3e^2=4$
$\Rightarrow e=\frac{2}{\sqrt{3}}$

Hence, option $\left(C\right)$ is correct.