The lengths of the intercepts made by any circle of the coordinate axes are equal if the centre lies on the line (s) represented by

Let the equation of any circle be x2+y2+2gx+2fy+c=0  ...(i) For intercept made by the circle on x-axis, put y=0  in (i)⇒  x2+2gx+c=0 If x1,x2  are roots of (ii), then length of the intercept on x-axis is|x1−x2|=(x1+x2)2−4x1x2=2g2−c Similarly length of the intercept of the y-axis is 2f2−c Since the lengths of these intercepts are equal g2−c=f2−c Therefore, centre lies on x2−y2=0