Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin,

$\frac{2 b^{2}}{a}=8$ and $2 a e=2 b$

$\Rightarrow \frac{\mathrm{b}}{\mathrm{a}}=\mathrm{e}$ and $1-\mathrm{e}^{2}=\mathrm{e}^{2} \Rightarrow \mathrm{e}=\frac{1}{\sqrt{2}}$

$\Rightarrow b=4 \sqrt{2}$ and $a=8$

so equation of ellipse is $\frac{x^{2}}{64}+\frac{y^{2}}{32}=1$