The ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{\mathrm{y}2}{{\mathrm{b}}^2}=1(\mathrm{b}>\mathrm{a})$ and the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ cut at right angles. If e is the eccentricity of the ellipse then $2{\mathrm{e}}^2==$

Given ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1(\mathrm{a}<\mathrm{b})$

diff w.r.to 'x' on both sides

$$\begin{gathered} \Rightarrow \frac{2\mathrm{x}}{{\mathrm{a}}^2}+\frac{2\mathrm{y}}{{\mathrm{b}}^2}\frac{\mathrm{dy}}{\mathrm{dx}}=0 \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-{\mathrm{b}}^2\mathrm{x}}{{\mathrm{a}}^2\mathrm{y}} \end{gathered}$$

Given parabola ${\mathrm{y}}^2=4\mathrm{ax}$

diff w.r.to 'x' on b.s

$\begin{array}{rcl}2\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}& =& 4\mathrm{a}\\ & \Rightarrow & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2\mathrm{a}}{\mathrm{y}}\end{array}$

Since given curves cut at right angles then tangents to the curves are perpendicular to each other.

Let ${\mathrm{m}}_1=$slope of tangents to ellipse at $({\mathrm{x}}_1,{\mathrm{y}}_1)$

             $=\frac{-{\mathrm{b}}^2{\mathrm{x}}_1}{{\mathrm{a}}^2{\mathrm{y}}_1}$

       ${\mathrm{m}}_2=$slope of tangent to parabola at $({\mathrm{x}}_1,{\mathrm{y}}_1)$

              $=\frac{2\mathrm{a}}{{\mathrm{y}}_1}$

Now

 $$\begin{gathered} {\mathrm{m}}_1\times {\mathrm{m}}_2=-1 \\ \Rightarrow \left(\frac{-{\mathrm{b}}^2{\mathrm{x}}_1}{{\mathrm{a}}^2{\mathrm{y}}_1}\right)\left(\frac{2\mathrm{a}}{{\mathrm{y}}_1}\right)=-1 \\ \Rightarrow 2{\mathrm{ab}}^2{\mathrm{x}}_1={\mathrm{a}}^2{{\mathrm{y}}_1}^2 \\ \Rightarrow 2{\mathrm{ab}}^2{\mathrm{x}}_1={\mathrm{a}}^2\left(4{\mathrm{ax}}_1\right)(\because ({\mathrm{x}}_1,{\mathrm{y}}_1) \mathrm{lies} \mathrm{on} {\mathrm{y}}^2=4\mathrm{ax}) \\ \Rightarrow \frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}=2 \\ \Rightarrow \frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}=\frac{1}{2} \end{gathered}$$

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

Now $2{\mathrm{e}}^2=2\left(\frac{1}{2}\right)=1$