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JEE Main 202523 JanuaryMorning Shift

Exam: JEE Main
Year: 2025
Date: 23 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Parabola

JEE Main 2025 · 23 January · Morning Shift · Mathematics · Parabola

If the line 3 x-2 y+12=0 intersects the parabola 4 y=3 x^2 at the...

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

A B

subtends an angle equal to

Options

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

Correct answer

B. $\tan ^{-1}\left(\frac{9}{7}\right)$

Step-by-step solution

\begin{aligned} & 3 x-2 y+12=0 \\ & 4 y=3 x^2 \\ & \therefore 2(3 x+12)=3 x^2 \\ & \Rightarrow x^2-2 x-8=0 \\ & \Rightarrow 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) \text{ (angle will be acute) }\end{aligned}

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