Question

Let $P \left( x _1, y _1\right)$ (where $y _1>0$ ) be any point on the parabola $y ^2=4 x$, with focus at $S$. Also normal drawn to the parabola at $P$ cuts the circle described on the focal radius of the point $P$ as diameter at $Q$. If length of $P Q$ is $\sqrt{10}$, then find the perpendicular distance of the point $P$ from the line $x+1=0$.

Answer 1

$l_{ PQ }=l_{ SN }$
$l_{ PQ }=\sqrt{1+ t ^2}=\sqrt{10} \Rightarrow t =3 \text { or }-3 \text { (rejected) } $
$P \equiv(9,6)$
Hence, perpendicular distance of the point $P$ from the line $x+1=0$ is $t^2+1=10$.