JEE Mathematics 2025 Question with Solution

Given: The line is $3x - 2y + 12 = 0$ and the parabola is $4y = 3x^2$.

Find: The angle subtended by chord AB at the vertex of the parabola.

Rewrite both equations in slope form:

and

At the intersection points,

Multiplying by $4$,

So,

Dividing by $3$,

Factorizing,

Hence, $x = 4$ or $x = -2$.

Now find the corresponding coordinates on the parabola:

and

Therefore, the points are $A(4,12)$ and $B(-2,3)$.

For the parabola $y = \frac{3}{4}x^2$, the vertex is $V(0,0)$.

Compute the slopes of $VA$ and $VB$:

The angle $\theta$ between the two lines is given by

Substituting,

Therefore, the line segment AB subtends an angle $\tan^{-1}\left(\frac{9}{7}\right)$ at the vertex. The correct option is D.

Given: The line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$.

Find: The angle subtended by segment AB at the vertex.

Substitute $y = \frac{3}{4}x^2$ into the line equation:

Multiplying by $2$,

Rearranging,

Hence, the intersection points are $A(4,12)$ and $B(-2,3)$, and the vertex is $V(0,0)$.

Using the lines from the vertex to the endpoints of the chord,

So the required angle is

Thus, the line segment AB subtends $\tan^{-1}\left(\frac{9}{7}\right)$ at the vertex, so the correct option is D.