$ \mathrm{y}^2=4 \lambda \mathrm{x}, \mathrm{P}(\lambda, 2 \lambda) $
Slope of the tangent to the parabola at point P
$ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{4 \lambda}{2 \mathrm{y}}=\frac{4 \lambda}{2 \mathrm{x} 2 \lambda}=1 $
Slope of the tangent to the ellipse at P
$ \frac{2 x}{a^2}+\frac{2 y y^{\prime}}{b^2}=0 $
As tangents are perpendicular $y^{\prime}=-1$
$ \begin{aligned} & \Rightarrow \frac{2 \lambda}{\mathrm{a}^2}-\frac{4 \lambda}{\mathrm{~b}^2}=0 \Rightarrow \frac{\mathrm{a}^2}{\mathrm{~b}^2}=\frac{1}{2} \\ & \Rightarrow \mathrm{e}=\sqrt{1-\frac{1}{2}}=\frac{1}{\sqrt{2}} \end{aligned} $