Let PQ be a focal chord of the parabola y^{2} | vedclass

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Question

$PQ$ is focal chord

$m _{P R} \cdot m_{P Q}=-1$

$\frac{2 t }{ t ^{2}-3} 	imes \frac{-2 / t }{\frac{1}{ t ^{2}}-3}=-1$

$\left( t ^{2}-1ight

Vedclass · Accepted Answer

$3+2 \sqrt{2}$

Vedclass · Answer

$PQ$ is focal chord

$m _{P R} \cdot m_{P Q}=-1$

$\frac{2 t }{ t ^{2}-3} 	imes \frac{-2 / t }{\frac{1}{ t ^{2}}-3}=-1$

$\left( t ^{2}-1ight)^{2}=0$

$\Rightarrow t =1$

$\Rightarrow P$ and $Q$ must be end point of latus rectum:

$\quad P (1,2)$ and $Q (1,-2)$

$	herefore \frac{2 b ^{2}}{ a }=4$ and $ae =1$

$\because$ We know that $b ^{2}= a ^{2}\left(1- e ^{2}ight)$

$	herefore a =1+\sqrt{2}$

$\because e ^{2}=1-\frac{ b ^{2}}{ a ^{2}}$

$	herefore e ^{2}=3-2 \sqrt{2}$

$\frac{1}{ e ^{2}}=3+2 \sqrt{2}$

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