Show that if one of the circle x2+y2+2gx+c=0 and x2+y2+2g1x+c=0 lies with in the other then gg1 and c are both positive.

If you are familiar with radical axis theory, then this is easy to do. The y-axis will be the radical axis, and since it is given that two of the circles are such that one is interior, that means the centres are either both on +ve x-axis or both on -ve x-axis. So gg₁>0. Similarly, the limiting point is real, which are  and  and so c>0.