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JEE Main 20252 AprilMorning Shift
Exam
JEE Main
Year
2025
Date
2 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Parabola

JEE Main 2025 · 2 April · Morning Shift · Mathematics · Parabola

Let the focal chord P Q of the parabola y^2=4 x make an angle of...

Let the focal chord PQP Q of the parabola y2=4xy^2=4 x make an angle of 60∘60^{\circ} with the positive xx-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 yy-axis at the point (0,α)(0, \alpha), then 5α25 \alpha^2 is equal to :Let the focal chord P Q of the parabola y^2=4 x make an angle of 60^ with the positive x -axis, where P lies in the

Options

  1. A15
  2. B25
  3. C30
  4. D20

Correct answer

A. 15

Step-by-step solution

Let the focal chord P Q of the parabola y^2=4 x make an angle of 60^ with the positive x -axis, where P lies in the
tan⁡60∘=2t−0t2−1=3⇒t=3\tan 60^{\circ}=\frac{2 \mathrm{t}-0}{\mathrm{t}^2-1}=\sqrt{3} \Rightarrow \mathrm{t}=\sqrt{3}
∴P(3,23)\therefore \mathrm{P}(3,2 \sqrt{3})
Circle :
(x−1)(x−3)+(y−0)(y−23)=0(x-1)(x-3)+(y-0)(y-2 \sqrt{3})=0
at x=0x=0
⇒3+y2−23y=0\Rightarrow 3+y^2-2 \sqrt{3} y=0
⇒y=3=α\Rightarrow \mathrm{y}=\sqrt{3}=\alpha
5α2=155 \alpha^2=15
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