Mathematics · Parabola

JEE Main 2024 — 9 April, Shift 2 — Question 21

Consider the circle $C: x^{2}+y^{2}=4$ and the parabola $P: y^{2}=8 x$ .If the set of all values of $\alpha$ , for which three chords of the circle C on three distinct lines passing through the point $(\alpha, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$ , then $(2 q-p)^{2}$ is equal to $\qquad$

Answer: 80

Numerical answer — enter this value.

Step-by-step solution

$\mathrm{T}=\mathrm{S}_{1}$

$\mathrm{xx}_{1}+\mathrm{yy}_{1}=\mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}$

$\alpha \mathrm{x}_{1}=\mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}$

$\alpha\left(2 \mathrm{t}^{2}\right)=4 \mathrm{t}^{4}+16 \mathrm{t}^{2}$

$\alpha=2 \mathrm{t}^{2}+8$

$\frac{\alpha-8}{2}=\mathrm{t}^{2}$

Also, $4 \mathrm{t}^{4}+16 \mathrm{t}^{2}-4<0$

$t^{2}=-2+\sqrt{5}$

$\alpha=4+2 \sqrt{5}$

$\therefore \alpha \in(8,4+2 \sqrt{5})$

$\therefore(2 \mathrm{q}-\mathrm{p})^{2}=80$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 9 April, Shift 2
Subject: Mathematics
Chapter: Parabola
Topic: Introduction to Parabola

Let A, B and C be three points on the parabola y^2=6 x and let the line segment A B meet the line L through C parallel to the x -axis at…

Answer: 80

Step-by-step solution

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