Question

Let $E$ denote the parabola $y^2=8x.\text{Let}P=(-2,4)$ 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 - angled 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)

$F$ lies on the line joining $Q$ and $Q^{\prime }$

$E:y^2=8x$
$P:(-2,4)$


Point $P(-2,4)$ lies on directrix $(x=-2)$ of parabola $y^2=8x$

So, $\angle QP{Q}^{\prime }=\frac{\pi }{2}$ and chord $Q{Q}^{\prime }$ is a focal chord and segment $PQ$ subtends right angle at the focus.

Chord of contact $\left(Q{Q}^{\prime }\right)T=0$
$\Rightarrow 4y=8\times \frac{x-2}{2}$
$\Rightarrow y=x-2$
Clearly, $F(2,0)$ lies on $Q{Q}^{\prime }$
$PF=\sqrt{(2+2{)}^2+4^2}=4\sqrt{2}$