The locus of the centre of the circle which cuts the circles x2+y2+2g1x+2f1y+c1=0 and x2+y2+2g2x+2f2y+c2=0 orthogonally, is

Let the circle be x2+y2+2gx+2fy+c=0This circle cuts the two given circles orthogonally∴ 2gg1+ff1=c+c1and 2gg2+ff2=c+c2Subtracting (ii) from (i), we get2gg1−g2+2ff1−f2=c1−c2.−2xg1−g2−2yf1−f2=c1−c2⇒ 2xg1−g2+2yf1−f2+c1−c2=0