Equation of chord of circle $S\equiv x^2+y^2+2gx+2fy+c=0$ in terms of the mid point $M\left(x_1,y_1\right)$ is $y-y_1=-\frac{x_1+g}{y_1+f}\left(x-x_1\right)$

which is designated by $T=S_1$.

Now compare the given equation of circle with $S\equiv x^2+y^2+2gx+2fy+c=0$ we get

$g=-1,f=0$

Now,

$y-y_1=-\frac{x_1+g}{y_1+f}\left(x-x_1\right)$

$-y_1=-\frac{x_1-1}{y_1}\left(-x_1\right)$

$-{y_1}^2={x_1}^2-x_1$

Now replace $\left(x_1,y_1\right)$by $\left(x,y\right)$ we get

$x^2+y^2-x=0$