Mathematics · Parabola

JEE Main 2025 — 23 January, Morning Shift — Question 4

If the line $3 x-2 y+12=0$ intersects the parabola $4 y=3 x^{2}$ at the points $A$ and $B$ , then at the vertex of the parabola, the line segment AB subtends an angle equal to

Option A:
$\tan ^{-1}\left(\frac{11}{9}\right)$
Option B:
$\frac{\pi}{2}-\tan ^{-1}\left(\frac{3}{2}\right)$
Option C:
$\tan ^{-1}\left(\frac{4}{5}\right)$
Option D:
$\tan ^{-1}\left(\frac{9}{7}\right)$
Correct

Answer: D

Step-by-step solution

$3 x-2 y+12=0$

$4 y=3 x^{2}$

$\therefore 2(3 \mathrm{x}+12)=3 \mathrm{x}^{2}$

$\Rightarrow \mathrm{x}^{2}-2 \mathrm{x}-8=0$

$\Rightarrow \mathrm{x}=-2,4$

$\mathrm{m}_{\mathrm{OA}}=-3 / 2, \mathrm{~m}_{\mathrm{OB}}=3$

$\tan \theta=\left(\frac{\frac{-3}{2}-3}{1-\frac{9}{2}}\right)=\frac{9}{7}$

$\theta=\tan ^{-1}\left(\frac{9}{7}\right)$ (angle will be acute)

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 23 January, Morning Shift
Subject: Mathematics
Chapter: Parabola
Topic: Considering a Line or a Point wrt a Parabola