The area of an ellipse is $8 \mathrm{\pi }$ sq. units  and distance  between the foci is $4\sqrt{3}$ then e =

$$\begin{gathered} Given taht \mathrm{\pi ab}=8\mathrm{\pi } \\ \mathrm{ab}=8 \\ \mathrm{distance} \mathrm{between} \mathrm{foci}=2\mathrm{ae}=4\sqrt{3} \\ \Rightarrow \mathrm{ae}=2\sqrt{3} \\ Now \mathrm{ab}=8 \Rightarrow {\mathrm{a}}^2{\mathrm{b}}^2=64 \\ \Rightarrow {\mathrm{a}}^2 {\mathrm{a}}^2(1-{\mathrm{e}}^2)=64 \\ {\mathrm{a}}^2({\mathrm{a}}^2-12)=64 \\ {\mathrm{a}}^4-12{\mathrm{a}}^2-64=0 \\ ({\mathrm{a}}^2-16) ({\mathrm{a}}^2+4)=0 \\ {\mathrm{a}}^2=16 \Rightarrow \mathrm{a}=4 \\ \mathrm{ae}=2\sqrt{3} \Rightarrow \mathrm{e}=\raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \end{gathered}$$