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

The correct option is B $\frac{2}{\sqrt{3}}$
length of latus rectum$=8$ ....Given

So, $\frac{2b^2}{a}=8$

Length of conjugate axis = Half of distance between focii

So, $2b=ae$

$\therefore e=\frac{2b}{a}$

$\frac{2b}{a}\times b=8$

$e\times b=8$ ; $b=\frac{8}{e}$

$a=\frac{2b}{e}=\frac{16}{e^2}$

$e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac{\frac{64}{e^2}}{\frac{16\times 16}{e^4}}}$

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

$e^2=1+\frac{e^2}{4}$

$4e^2=4+e^2$

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

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