If the distance between the foci of an ellipse is equal to the...

Given, In ellipse the distance between the foci = Length of the latusrectum $$\begin{array}{ll}\Rightarrow \quad & \text { 2ae }=\frac{2 b^{2}}{a} \\ \Rightarrow \quad & \quad a^{2} e=b^{2} \Rightarrow e=\frac{b^{2}}{a^{2}}\end{array}$$\because \quad e^{2}=1-$\frac{b^{2}}{a^{2}}\Rightarrow \quad$e^{2}=1-e \quad$[from Eq.$\Rightarrow e^{2}+e-1=0\Rightarrow \quad e=$\frac{-1 \pm \sqrt{1+4}}{2} (by quadratic formula)$\Rightarrow \quad $e=\frac{-1 \pm \sqrt{5}}{2}\Rightarrowe=\frac{\sqrt{5}-1}{2} \quad(\because1>e>0)$