The tangent at point $p(x_1,y_1)$to the parabola $y^2=4ax$meets the parabola $y^2=4a(x+b)$at Q and K, the coordinates of the mid -point of QK are

Equation of the tangent at the point $p(x_1,y_1)$

To the parabola$y^2-4ax$ is $y{y}_1=2a(x+x_1)$

(Or)

$2ax-y_{1}y+2a{x}_1=0\to \left(1\right)$

If $M(h,k)$is the mid -point of QK. Then the equation of QK. A chord of the parabola 

$y^2=4a(x+b)$, in terms of its mid point is $ky-2a(x+h)-4ab=k^2-4a(h+b)$

Using $T=S^1$$2ax-ky+k^2=2ah=0\to \left(2\right)$

Since us and (2) represents same line we have

$\frac{2a}{2a}=\frac{y_1}{k}=\frac{2a{x}_1}{k^2-2ah}$

$\Rightarrow k=y_{1}and k^2-2ah=2a{x}_1$

$$\begin{gathered} y_1^2-2ah=2a{x}_1\Rightarrow 4a{x}_1-2a{x}_1=2ah \\ \Rightarrow h=x_1 \\ \therefore \left(h_{1}K\right)=\left(x_1{y}_1\right) \end{gathered}$$