Question

$PQ$ is a normal chord of the parabola $y^2=4x$ 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 $P\left(a_1^2,2a_1\right)$
Relation between $t_1\&t_2$
$t_2=-t_1-\frac{2}{t_1}$
equation of line $PR$
$y-2a{t}_1=\frac{2}{t_2}\left(x-a{t}_1^2\right)$
Put $y=0$ and $t_2=-t_1-\frac{2}{t_1}$, we get
$R=\left(\left(-a{t}_1{t}_2+a{t}_1^2\right),0\right)$
$R=\left(2a\left(1+t_1^2\right),0\right)$
Length of $PS=a\left(1+t_1^2\right)$
So AR is twice of PS.