Circles are drawn through the point ( 3, 0) to cut an intercept of length 6 units on the negative direction of the x-axis. The equation of the locus of their centres, is

Let the equation of the circle bex2+y2+2gx+2fy+c=0It passes through ( 3, 0). Therefore9+6g+c=0Circle (i) cuts an intercept of 6 units on the negative direction of x-axis.∴ 2g2−c=6⇒g2−c=9From (ii) and (iii), we get9+6g+g2=9⇒ g(g+6)=0⇒g=0 [∵g>0∴g+6≠0]Hence, the locus of ( - g, - f ) is - x = 0, i.e. x = 0.