Let S denote the locus of the point of intersection of the pair of lines begin gathered 4 x-3 y=12 α, \\ 4 α x+3 α y=12, end gathered where α varies over the set of non-zero real numbers. Let T be the tangent to S passing through the points (p, 0) and (0, q), q 0 , and parallel to the line 4 x-frac 3 √(2) y=0 . Then the value of p q is

Mathematics · Hyperbola

JEE Advanced 2025 — Paper 2 — Question 20

Let $S$ denote the locus of the point of intersection of the pair of lines $\begin{gathered} 4 x-3 y=12 \alpha, \\ 4 \alpha x+3 \alpha y=12, \end{gathered}$ where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to

$S$ passing through the points $(p, 0)$ and $(0, q), q>0$ , and parallel to the line $4 x-\frac{3}{\sqrt{2}} y=0$ . Then the value of $p q$ is

Option A:
$-6 \sqrt{2}$
Correct
Option B:
$-3 \sqrt{2}$
Option C:
$-9 \sqrt{2}$
Option D:
$-12 \sqrt{2}$

Answer: A

Step-by-step solution

$\left.\quad \begin{array}{rl}4 x-3 y & =12 \alpha \\ 4 x+3 y & =\frac{12}{\alpha}\end{array}\right\} \rightarrow 16 x^{2}-9 y^{2}=144$

$\frac{x^{2}}{9}-\frac{y^{2}}{16}=1 \rightarrow$ Curve : $S$

$T: y=m x \pm \sqrt{9 m^{2}-16}$

$\mathrm{m}=\frac{4 \sqrt{2}}{3}$

$y=\frac{4 \sqrt{2} x}{3} \pm \sqrt{32-16}$

$3 y=4 \sqrt{2} x \pm 12$

as $\mathrm{q}>0$

$3 y=4 \sqrt{2} x+12$

$\mathrm{p}=-\frac{3}{\sqrt{2}} \quad \& \mathrm{q}=4$

$\mathrm{pq}=-6 \sqrt{2}$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2025
Paper: Paper 2
Subject: Mathematics
Chapter: Hyperbola
Topic: Introduction to Hyperbola