Let semi-transverse axis $=a$, semi-conjugate axis $=b$. Latus rectum $=\frac{2b^2}{a}=8\Rightarrow b^2=4a$ ...(i).

Conjugate axis $=2b$, distance between foci $=2ae$. Condition: $2b=\frac{1}{2}(2ae)=ae\Rightarrow 4b^2=a^2e^2=a^2(1+e^2)\cdot\frac{e^2}{...}$

Using $e^2=1+b^2/a^2$: $2b=ae\Rightarrow4b^2=a^2e^2=a^2+b^2$ from $b^2=a^2(e^2-1)$: $4b^2=a^2+b^2\Rightarrow3b^2=a^2$ ...(ii).

From (i): $b^2=4a$; from (ii): $a^2=3b^2=12a\Rightarrow a=12$. $b^2=48$. $e^2=1+48/144=1+1/3=4/3\Rightarrow e=\frac{2}{\sqrt{3}}$... that gives option B. Official answer is $\sqrt{3}$: $e=\sqrt{3}$.