Question

A circle passes through the origin and has its centre on $y=x$. If it cuts $x^2+y^2-4x-6y+10=0$ orthogonally, then show that the equation of the circle is $x^2+y^2-2x-2y=0$.

Answer 1

Since the circle passes through origin $c=0$, its centre lies on $y=x\therefore g=f$. It cuts orthogonally the given circle
$\therefore 2g(-2)+2f(-3)=10+0$ But $g=f$
$\therefore g=-1=f$ and $c=0$.