Consider the circle C: x^2+y^2=4 and the parabola P: y^2=8 x . If the set of all values of α , for which three chords of the circle C on three distinct lines passing through the point (α, 0) are bisected by the parabola

Correct answer: 0 aligned & T=S_1 & x_1+y_1=x_1^2+y_1^2 & α x_1=x_1^2+y_1^2 & α(2 t^2)=4 t^4+16 t^2 & α=2 t^2+8 & (α-8)/(2)=t^2 aligned Also, 4 t^4+16 t^2-4 < 0 aligned & t^2=-2+ 5 & α=4+2 5 & α ∈(8,4+2 5 ) & (2 q-p)^2=80 aligned

JEE Main 20249 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 9 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Parabola

JEE Main 2024 · 9 April · Evening Shift · Mathematics · Parabola

Consider the circle C: x^2+y^2=4 and the parabola P: y^2=8 x . If...

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 ________ Consider the circle C: x^2+y^2=4 and the parabola P: y^2=8 x . If the set of all values of α , for which three chords

Correct answer

Step-by-step solution

\begin{aligned} & T=S_1 \\ & x_1+y_1=x_1^2+y_1^2 \\ & \alpha x_1=x_1^2+y_1^2 \\ & \alpha\left(2 t^2\right)=4 t^4+16 t^2 \\ & \alpha=2 t^2+8 \\ & \frac{\alpha-8}{2}=t^2 \end{aligned}

Also,

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

\begin{aligned} & \mathrm{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 \end{aligned}

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