Question

Let $E$ denote the parabola $y^2=8x$. Let $P=(-2,4)$ and let $Q$ and $Q^{\prime }$ be two distinct points on $E$ such that the lines $PQ$ and $P{Q}^{\prime }$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is(are) TRUE?

• A. The triangle $PFQ$ is a right-angled triangle
• B. The triangle $QP{Q}^{\prime }$ is a right-angle triangle
• C. The distance between $P$ and $F$ is $5\sqrt{2}$
• D. $F$ lies on the line joining $Q$ and $Q^{\prime }$

Answer 1

Answer: (A), (B), (D)

$E=y^2-8x=0$
$a=2$
Let $Q\left(2t_1^2,4t_1\right),Q^{\prime }\left(2t_2^2,4t_2\right)$
$t_1{t}_2=-1,t_1+t_2=2$
$t_1=1+\sqrt{2},t_2=1-\sqrt{2}$
(A) ( slope of $PF$)( slope of $FQ)=-1$
$\Rightarrow \angle PFQ=\frac{\pi }{2}$
(B) ( slope of $P{Q}^{\prime })($ slope of $PQ)=-1$
$\angle QP{Q}^{\prime }=\frac{\pi }{2}$
(C) $PF=4\sqrt{2}$
(D) slope of $Q^{\prime }F=$ slope of $FQ$
$\Rightarrow Q,F,Q^{\prime }$ are collinear