A uniform lamina \(O A B C D\) is in the form of a rectangle, \(O B C D\), joined along the edge \(O B\) to a quarter circle \(O A B\). The length of \(D O\) is \(k a\) and the length of \(O B\) is \(a\). The lamina rests in a vertical plane with its edge \(C B\) on a horizontal surface (see diagram). (a) Find, in terms of \(k, a\) and \(\pi\), an expression for the distance of the centre of mass above the horizontal surface. [You may use without proof the result for the centre of mass of a circular sector in the list of formulae (MF19).]

The lamina is on the point of toppling about \(B\). (b) Find the value of \(k\).