When the mass
kM is added onto the lamina, the overall centre of mass moves. We are told that when the system S is freely suspended from A, the line OA is the vertical. Now you need to know that the centre of mass of S must be on the vertical through, so it must be on the line through O and A.
This means you have this situation as shown below. The blue force lines are parallel to the green line OA, as mass acts parallel to it when hung in equilibrium. As the CoM is on the green line, and equilibrium is obtained, this means clockwise and anticlockwise moments on either side of the green line must cancel each other out. The two moments here in question are the forces from the two centres of mass.
Does this make more sense, and can you see how you can finish off the question by considering moments?