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 $$(\alpha, 0)$$

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