JEE Main 2026 Apr 6 Shift 1, Mathematics Q12: Parabola: Chord, Tangent and Normal

JEE Main2026Apr 6, Shift 1Mathematics

Let chord PQ of length $313$ of the parabola $y^{2} = 12 x$ be such that the ordinates of points P and Q are in the ratio $1 : 2$ . If the chord PQ subtends an angle $α$ at the focus of the parabola, then $sin α$ is equal to:

A
$\frac{3}{5}$
B
$\frac{4}{5}$
C
$\frac{5}{13}$
D
$\frac{12}{13}$

Solution

For $y^{2} = 12 x$ , use the parametrization $(3 t^{2}, 6 t)$ . Take $P = (3 t^{2}, 6 t)$ and, to keep the ordinate ratio $1 : 2$ , $Q = (12 t^{2}, 12 t)$ .

Then $P Q^{2} = (12 t^{2} - 3 t^{2})^{2} + (12 t - 6 t)^{2} = 81 t^{4} + 36 t^{2} = (313)^{2} = 117$ .

So $9 t^{4} + 4 t^{2} - 13 = 0$ , factoring as $(t^{2} - 1) (9 t^{2} + 13) = 0$ , giving $t = 1$ .

Hence $P = (3, 6)$ and $Q = (12, 12)$ . The focus is $S = (3, 0)$ , so $S P$ is vertical and $S Q$ has slope $\frac{12 - 0}{12 - 3} = \frac{4}{3}$ .

The angle $α$ between $S P$ (vertical) and $S Q$ satisfies $tan α = \frac{3}{4}$ , hence $sin α = \frac{3}{5}$ .

Concept behind this question

Parabola: Chord, Tangent and Normal Notes, formulas and examples →

Practice more Mathematics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →

JEE Main 2026 Apr 6 Shift 1, Mathematics Q12: Parabola: Chord, Tangent and Normal

Concept behind this question

More previous year questions on Parabola: Chord, Tangent and Normal

Practice more Mathematics