Answer : $\sqrt 5$
The latus rectum of the hyperbola is $8$
It is given that the conjugate axis is equal to $\large\frac{1}{2}$ of the distance between the foci.
Length of the latus rectum for a hyperbola is $\large\frac{2a^2}{b^2}$$=8$
$\Rightarrow 2a^2=8b^2$
$a^2=4b^2$
$a=2b$
We know that $a^2=b^2(e^2-1)$
$4b^2=b^2(e^2-1)$
$\Rightarrow e^2=5$
$e=\sqrt 5$