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…

Question

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 ________

Aicharya · Accepted Answer

$\begin{aligned} & T=S_1 \ & x_1+y_1=x_1^2+y_1^2 \ & \alpha x_1=x_1^2+y_1^2 \ & \alpha\left(2 t^2ight)=4 t^4+16 t^2 \ & \alpha=2 t^2+8 \ & \frac{\alpha-8}{2}=t^2 \end{aligned}$ Also, $4 \mathrm{t}^4+16 \mathrm{t}^2-4 < 0$ $\begin{aligned} & \mathrm{t}^2=-2+\sqrt{5} \ & \alpha=4+2 \sqrt{5} \ & 	herefore \alpha \in(8,4+2 \sqrt{5}) \ & 	herefore(2 \mathrm{q}-\mathrm{p})^2=80 \end{aligned}$

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…

Solution