Question

Consider the parabola $y^2=4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P=(-2,1)$ meet the parabola at $P_1$ and $P_2$. Let $Q_1$ and $Q_2$ be points on the lines $S P_1$ and $S P_2$ respectively such that $P Q_1$ is perpendicular to $S P_1$ and $P Q_2$ is perpendicular to $S P_2$. Then, which of the following is/are TRUE ?

• A. $S Q_l=2$
• B. $Q _1 Q _2=\frac{3 \sqrt{10}}{5}$
• C. $PQ _1=3$
• D. $SQ _2=1$

Answer 1

Answer: (B), (C), (D)

Let equation of tangent with slope ' $m$ ' be

$T : y = mx +\frac{1}{ m }$
$T$ : passes through $(-2,1)$ so
$1=-2 m+\frac{1}{m}$
$\Rightarrow m =-1 \text { or } m =\frac{1}{2}$
Points are given by $\left(\frac{ a }{ m ^2}, \frac{2 a }{ m }\right)$
So, one point will be $(1,-2) \&(4,4)$
Let $P_1(4,4) \& P_2(1,-2)$
$ P _1 S : 4 x -3 y -4=0$
$ P _2 S : x -1=0$
$ PQ _1=\left|\frac{4(-2)-3(1)-4}{5}\right|=3 $
$ SP =\sqrt{10} ; PQ _2=3 ; SQ _1=1= SQ _2 $
$ \frac{1}{2}\left(\frac{ Q _1 Q _2}{2}\right) \times \sqrt{10}=\frac{1}{2} \times 3 \times 1 \text { (comparing Areas) }$
$\Rightarrow Q _1 Q _2=\frac{2 \times 3}{\sqrt{10}}=\frac{3 \sqrt{10}}{5}$