Let $E$ denote the parabola $y^{2} = 8 x$ . Let $P = (- 2, 4)$ ) and let $Q$ and $Q^{1}$ be two distinct points on $E$ such that the lines $P Q$ and $P Q^{1}$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) FALSE?

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The distance between $P$ and $F$ is 5 $\sqrt{2}$

F lies on the line joining $Q$ and $Q^{1}$

The triangle $Q P Q^{1}$ is an obtuse angled triangle

The triangle $P F Q$ is a right-angled triangle

answer is B, C.

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Detailed Solution

E = y² - 8x = 0 a = 2
Let $Q (2 t_{1}^{2}, 4 t_{1}), Q^{'} (2 t_{2}^{2}, 4 t_{2}) t_{1} t_{2} = 1 + \sqrt{2}, t_{2} = 1 - \sqrt{2}$

A) (Slope of PF) (Slope of FQ) = -1
$\Rightarrow ∠ PFQ = \frac{π}{2}$
B) (Slope of PQ') (Slope of PQ) = -1
$\Rightarrow ∠ {QPQ}^{'} = \frac{π}{2}$
C) PF = 4 $\sqrt{2}$
D) Slope of Q'F = Slope of FQ
$\Rightarrow$ Q, F, Q' are collinear