Let a , b and lambda be positive real numbers | vedclass

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Question

$y ^2=4 \lambda x , P (\lambda, 2 \lambda)$

Slope of the tangent to the parabola at point $P$

$\frac{ dy }{ dx }=\frac{4 \lambda}{2 y }=\frac{4

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}}$

Vedclass · Answer

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

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