Let S and S ' be the foci of an ellipse and B be any one of the extremities of its minor axis. If Δ S '…

Given that, 

$\angle SB{S}^{\text{'}}=\frac{\pi }{2}$

$\Rightarrow \angle OBS=\frac{\pi }{4}$

$\Rightarrow b=ae ...\left(i\right)$ 

Also, area $\left(∆SB{S}^{\text{'}}\right)=8$ 

$\Rightarrow \frac{1}{2}.2ae.b=8$

$\Rightarrow aeb=8 ...\left(ii\right)$ 

From equations $\left(i\right) \& \left(ii\right)$, 

$b^2=8$ and $a^2=b^2+a^2{e}^2=8+8=16$

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

Thus, the length of the latus rectum is $\frac{2b^2}{a}=\frac{16}{4}=4$