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JEE 2025 Mathematics PYQ — Let the focal chord PQ of the parabola y 2 = 4x make an angle of … | Mathem Solvex | Mathem Solvex

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JEE 2025 — Mathematics PYQ

JEE | Mathematics | 2025

Let the focal chord PQ of the parabola y² = 4x make an angle of 60° with the positive
x-axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y-axis at the point (0, α), then 5α² is equal to:

Choose the correct answer:

A.
15
(Correct Answer)
B.
25
C.
30
D.
20

Correct Answer:

Explanation

Parabola $\Rightarrow y^{2} = 4 x$
Let, parametric coordinate of $P = (t^{2}, 2 t)$
Focus $(S) = (1, 0)$
$tan 6 0^{\circ} = \frac{2 t - 0}{t ^{2} - 1} = 3 \Rightarrow t = 3$
$∴ P (3, 23)$
Equation of circle in diameter for $m$ is:
$(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$
$(x - 1) (x - 3) + (y - 0) (y - 23) = 0$
at $x = 0 \Rightarrow 3 + y^{2} - 23 y = 0$
$\Rightarrow y = 3 = α$
$5 α^{2} = 15$

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