Parabola
Chords of Circle Bisected by Parabola — Interval
nta_pyq_2024_apr
Grade 11
Question:
Consider the circle $C:x^2+y^2=4$ and the parabola $P:y^2=8x$. 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 $(2q-p)^2$ is equal to
Step-by-Step Solution
Key Concept: Let midpoint of a chord of $C$ be $(x_1,y_1)=(2t^2,4t)$ on the parabola $y^2=8x$. By $T=S_1$: chord of circle has equation $x\cdot2t^2+y\cdot4t=4t^4+16t^2$. This passes through $(\alpha,0)$: $\alpha\cdot2t^2=4t^4+16t^2\Rightarrow\alpha=2t^2+8$.
$\alpha\in(8,4+2\sqrt{5})$. $(2q-p)^2=80$.
Correct Answer: 80