The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance…

For any standard hyperbola we know that

eccentricity, $e=\sqrt{1+\frac{b^2}{a^2}}$, conjugate axis $=2b$ and distance between the foci $=2ae$.

Given,  $2b=\frac{1}{2}\left(2ae\right)$. 

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

Now, $e= \sqrt{1+\frac{b^2}{a^2}}= \sqrt{1+\frac{e^2}{4}}$

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

$\Rightarrow \frac{3}{4}{e}^2=1$

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

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