Find the equation to the locus of the middle point of the chord of the circle $x^2+y^2+2gx+2fy+c=0$ which subtends right angle at a given point $(a,b)$
My attempt is as follows:-
Equation of chord passing through the midpoint $(h,k)$
$$xh+yk+g(x+h)+k(y+f)+c=h^2+k^2+2hg+2fk+c$$ $$x(h+g)+y(k+f)=h^2+k^2+hg+fk$$
$$x=\dfrac{-y(k+f)+h^2+k^2+hg+fk}{h+g}\tag{1}$$
Now we if we place $x$ in the equation of circle to find endpoints of chord, equation will become very huge, so how to proceed from here?