Question

If a circle passes through the point $(a,b)$ and cuts the circle $x^2+y^2=k^2$ orthogonally. Find the equation of the locus of its centre.

Answer 1

Let the equation of the circle is

$x^2+y^2+2lx+2my+c=0⟶\left(1\right)$

where (l,m) be its centre

also, the equation of another circle is

$x^2+y^2=k^2\Rightarrow x^2+y^2+2x.0+2y.0-k^2=0⟶\left(2\right)$

since circle(1) cuts circle (2) orthogonally then,

$2g{g}^{\prime }+2f{f}^{\prime }=c+c^{\prime }\Rightarrow 2.l.0+2.m.0=c-k^2\Rightarrow c=k^2$

also circle (1)passes through (a,b), so the equation(1) be like

$a^2+b^2+2al+2bm+k^2=0$

therefore locus of the centre is

$a^2+b^2+2ax+2by+k^2=0$