If a circle passes through the point $(a, b)$ and cuts the circle $x^2+y^2=4$ orthogonally, then the locus…
If a circle passes through the point $(a, b)$ and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its centre is
$2 a x+2 b y+\left(a^2+b^2+4\right)=0$
$2 a x+2 b y-\left(a^2+b^2+4\right)=0$
$2 a x-2 b y+\left(a^2+b^2+4\right)=0$
$2 a x-2 b y-\left(a^2+b^2+4\right)=0$
Solution
Let the circle be $x^2+y^2+2 g x+2 f y+c=0 \Rightarrow c=4$ and it passes through $(a, b)$ $\Rightarrow \mathrm{a}^2+\mathrm{b}^2+2 \mathrm{ga}+2 \mathrm{fb}+4=0$
Hence locus of the centre is $2 a x+2 b y-\left(a^2+b^2+4\right)=0$