JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Parabola

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:

Answer: (A)

Solution

Points on

y^{2} = 12 x

have the form

(3 t^{2}, 6 t)

. Taking

P = (3 t^{2}, 6 t)

and

Q = (3 s^{2}, 6 s)

with ordinates in ratio

1 : 2

gives

s = 2 t

, so

Q = (12 t^{2}, 12 t)

. Then

P Q = (9 t^{2})^{2} + (6 t)^{2} = 3 t 9 t^{2} + 4

, and

P Q = 313

gives

9 t^{4} + 4 t^{2} - 13 = 0

, i.e.

(t^{2} - 1) (9 t^{2} + 13) = 0

, so

t = 1

. Hence

P = (3, 6)

Q = (12, 12)

and the focus is

S = (3, 0)

. The segment

S P

is vertical and

S Q

has slope

\frac{12}{12 - 3} = \frac{4}{3}

, so the angle

α

between them satisfies

tan α = \frac{4}{3}

, giving

sin α = \frac{3}{5}

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Single correct option