If circle $x^2$ +$ y^2$ +2gx +2fy -12 = 0 is orthogonal to the circle $x^2$ + $y^2$ -4x -6y -2 = 0, the find the locus of the centre of the first circle
I have tried using the relation for orthogonal circles : ${r_1}^2$ + ${r_2}^2$ = $d^2$ where d is the distance between the centres of the circle.
${r_2}$ and $C_2$ can be found since the second equation of circle is known completely.
${r_2}$ = √15
$C_2$ : (2,3)
From there on I am lost what to do next. Can anyone give a hint to solve the problem?
Assuming the center of $C_1$ to be $(-g, -f)$, we can write $(2 + g)^2 + (3 + f)^2 = 15 + g^2 + f^2 + 12$. Replace $g, f$ by $x, y$ and that is the locus of the center of the circle.