The locus of midpoints of chords of the circle $x^2+y^2-2px=0,$ passing through the origin is 

Let $P\left(x_1, y_1\right)$ be a point on the locus.

Equation of the chord of the circle $x^2+y^2-2px=0$having $P\left(x_1, y_1\right)$ as midpoint is $S_1=S_{11}$

$\Rightarrow x{x}_1+y{y}_1-p\left(x+x_1\right)=$$x_1^2+y_1^2-2p{x}_1\Rightarrow \left(x_1-p\right)x+\left(y_1\right)y=x_1^2+y_1^2-p{x}_1$

This passes through the origin $\left(0, 0\right)\Rightarrow \left(x_1-p\right)\left(0\right)+y_1\left(0\right)=$$x_1^2+y_1^2-p{x}_1\Rightarrow x_1^2+y_1^2-p{x}_1=0$

Locus of $\left(x_1, y_1\right)$ is $x^2+y^2-px=0$.