Mathematics - Circle Question with Solution | TestHub

Let the mid-point be $\left(x_1,y_1\right)$. The chord is $T=S_1$ : $x{x}_1+y{y}_1+g\left(x+x_1\right)+f\left(y+y_1\right)+c=x_1^2+y_1^2+2g{x}_1+2f{y}_1+c$. It passes through $(0,0)$, so $g{x}_1+f{y}_1=x_1^2+y_1^2+2g{x}_1+2f{y}_1$, i.e. $x_1^2+y_1^2+g{x}_1+f{y}_1=0$. The locus is $x^2+y^2+gx+fy=0$.