Let the equation of the circle which passes through the point (a,b) is given by,
x2+y2+2gx+2fy+c=0..........(1)
a2+b2+2ga+2fb+c=0........(2)
Now circle (1) cuts the circle x2+y2=4 orthogonally
Therefore,
2g×0+2f×0=c−4
c=4.........(3)
Substituting c=4 in (2)
a2+b2+2ga+2fb+4=0........(2)
Now let the centre of the circle by (h,k)
therefore h=−g and k=−f
thus,
a2+b2−2ha−2kb+4=0
replace h by x and k by y,
thus the locus of the centre is
a2+b2−2xa−2yb+4=0
2(ax+by)−(a2+b2+4)=0