QJEEM24Mae6ac: JEE Main 2024 Mathematics - Class 11 JEEM24Mae6ac Solution: Consider the circle and the parabola . If the set of all values …

Chord with the middle point $(\alpha, 0)$

$\begin{aligned}& \Rightarrow T=S_1 \\& \Rightarrow y y_1-4\left(x+x_1\right)=y_1^2-8 x_1 \\& \Rightarrow-4(x+\alpha)=0-8 \alpha \\& \Rightarrow x+\alpha=2 \alpha \Rightarrow x=\alpha\end{aligned}$

For circle chord with $(2 t^2, 4 t)$ as mid point

$\begin{aligned}& \Rightarrow \quad T=S_1 \\& \Rightarrow \quad x x_1+y y_1-4=x_1^2+y_1^2-4 \\& \Rightarrow \quad 2 t^2 x+4 t y=4 t^4+16 t^2\end{aligned}$

Passes through $(\alpha, 0)$

$\begin{aligned}\Rightarrow & 2 t^2 \alpha=4 t^4+16 t^2 \\\Rightarrow & 2 \alpha=4 t^2+16 \Rightarrow \alpha=2 t^2+8=x_0+8 \\& x^2+y^2=4 \text { and } y^2=8 x \\\Rightarrow & x^2+8 x-4=0 \Rightarrow x_0=\frac{-8+\sqrt{80}}{2} \\\Rightarrow & p=8 \text { and } q=4+\frac{\sqrt{80}}{2} \Rightarrow(2 q-p)^2=80\end{aligned}$