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 PQ and PQ prime are tangents to E . Let F be the focus of E . Then which of the following statements is(are) TRUE?

The triangle PFQ is a right-angled triangle

The triangle QPQ' prime is a right-angle triangle

F lies on the line joining Q and Q prime

The distance between P and F is 5 √(2)

Mathematics · Parabola

JEE Advanced 2021 — Paper 2 — Question 22

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

Option A:
The triangle PFQ is a right-angled triangle
Correct
Option B:
The triangle QPQ' $^{\prime}$ is a right-angle triangle
Correct
Option C:
The distance between $P$ and $F$ is $5 \sqrt{2}$
Option D:
F lies on the line joining Q and $\mathrm{Q}^{\prime}$
Correct

Answer: A, B, D

Step-by-step solution

$E=y^{2}-8 x=0$

$\mathrm{a}=2$

Let $\mathrm{Q}\left(2 \mathrm{t}_{1}^{2}, 4 \mathrm{t}_{1}\right), \mathrm{Q}^{\prime}\left(2 \mathrm{t}_{2}^{2}, 4 \mathrm{t}_{2}\right)$

$\mathrm{t}_{1} \mathrm{t}_{2}=-1, \mathrm{t}_{1}+\mathrm{t}_{2}=2$

$t_{1}=1+\sqrt{2}, t_{2}=1-\sqrt{2}$

(A) (slope of PF) (slope of FQ) $=-1$

$\Rightarrow \angle \mathrm{PFQ}=\frac{\pi}{2}$

(B) $\left(\right.$ slope of $\left.\mathrm{PQ}^{\prime}\right)($ slope of PQ $)=-1$

$\angle \mathrm{QPQ}^{\prime}=\frac{\pi}{2}$

(D) slope of $Q^{\prime} F=$ slope of $F Q$

$\Rightarrow \mathrm{Q}, \mathrm{F}, \mathrm{Q}^{\prime}$ are collinear

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2021
Paper: Paper 2
Subject: Mathematics
Chapter: Parabola
Topic: Various form of tangents & normals, chord of contact