if a circle passes through (1,2) and cuts  $x^2+y^2=4,$ orthogonally  then the locus of centre is

$\begin{array}{l}{x}^2+y^2+2gx+2fy+C=0\\ orthogonalyto{x}^2+y^2=4thenC-4=0\\ and\left(1,2\right)\Rightarrow 1+4+2g+4f+4=0\\ 2g+4f+9=0\\ locus\left(-g,-f\right)=\left(x,y\right)\\ locusofcentreis2x+4y-9=0\end{array}$