If the circle $x^2+y^2+2gx+2fy+c=0$ bisects the circumference of the circle $x^2+y^2+2g^{\mathrm{\prime }}x+2f^{\mathrm{\prime }}y+c^{\mathrm{\prime }}=0$ then the length of the common chord of these two circles, is 

The length of the common chord of the two given circles is the length of the diameter of the circle  $x^2+y^2+2g^{\mathrm{\prime }}x+2f^{\mathrm{\prime }}y+c^{\mathrm{\prime }}=0$ 

Hence, the required length $=2\sqrt{g^{\mathrm{\prime }2}+f^{\mathrm{\prime }2}-c^{\mathrm{\prime }}}$