Question

$PQ$ is a normal chord of the parabola $y ^2=4 x$ at $P , A$ being the vertex of the parabola. Through $P$ a line is drawn parallel to $AQ$ meeting the $x$-axis in $R$. Then the length of of $AR$ is -

• A. equal to the length of the latus rectum
• B. equal to the focal distance of the point $P$.
• C. equal to twice the focal distance of the point $P$.
• D. equal to the distance of the point $P$ from the directrix

Answer 1

Answer: (C)

Let $P\left(a_1^2, 2 a_1\right)$
Relation between $t_1 \& t_2$
$t _2=- t _1-\frac{2}{ t _1}$
equation of line $PR$
$y-2 a t_1=\frac{2}{t_2}\left(x-a t_1^2\right)$
Put $y=0$ and $t_2=-t_1-\frac{2}{t_1}$, we get
$ R =\left(\left(- at _1 t _2+ at _1^2\right), 0\right) $
$ R =\left(2 a \left(1+ t _1^2\right), 0\right)$
Length of $P S=a\left(1+t_1^2\right)$
So AR is twice of PS.