$\begin{array}{1 1}(A)\;\frac{4}{3}\\(B)\;\frac{4}{\sqrt 3} \\(C)\;\frac{2}{\sqrt 3} \\(D)\;\text{none of these} \end{array}$

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We have $\large\frac{2b^2}{a}$$=8$

and $2b=\large\frac{1}{2}$$(2 ae)$

$\therefore \large\frac{2}{a}\bigg( \large\frac{ae}{2} \bigg)^2$$=8$

$ae^2=16$ -----(1)

Now $\large\frac{2b^2}{a}$$=8$

$b^2= 4a$

$a^2(e^2-1)=4a$

$ae^2-a=4$

Substitute the value of $ae^2=16$ in eq(2)

$16-a=4$

$-a=4-16$

$-a=-12$

$a=12$

$ae^2=16$

$e^2=\large\frac{16}{12}$

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

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

Hence C is the correct answer.

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