Question

Let $E$ denote the parabola $y ^{2}=8 x$. Let $P =(-2,4)$ and let $Q$ and $Q ^{\prime}$ be two distinct points on $E$ such that the lines $P Q$ 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 $QPQ '$ 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 $Q'$

Answer 1

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

$ E = y ^{2}-8 x =0 $
$ a =2$
Let $ Q \left(2 t _{1}^{2}, 4 t _{1}\right), Q '\left(2 t _{2}^{2}, 4 t _{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 $PQ')($ slope of $PQ )=-1$
$\angle QPQ'=\frac{\pi}{2}$
(C) $PF =4 \sqrt{2}$
(D) slope of $Q' F =$ slope of $FQ$
$\Rightarrow Q , F , Q'$ are collinear