A uniform lamina is in the form of a triangle \(A B C\) in which angle \(B\) is a right angle, \(A B=9 a\) and \(B C=6 a\). The point \(D\) is on \(B C\) such that \(B D=x\) (see diagram). The region \(A B D\) is removed from the lamina. The resulting shape \(A D C\) is placed with the edge \(D C\) on a horizontal surface and the plane \(A D C\) is vertical.

Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium.