A uniform lamina is in the form of an isosceles triangle \(ABC\), in which \(AC=2a\) and \(\angle ABC=90^\circ\). The point \(D\) on \(AB\) is such that the ratio \(DB:AB=1:k\). The point \(E\) on \(CB\) is such that \(DE\) is parallel to \(AC\). The triangle \(DBE\) is removed from the lamina.

(a) Find, in terms of \(k\), the distance of the centre of mass of the remaining lamina \(ADEC\) from the midpoint of \(AC\).

When the lamina \(ADEC\) is freely suspended from the vertex \(A\), the edge \(AC\) makes an angle \(\theta\) with the downward vertical, where \(\tan\theta=\dfrac5{18}\).

(b) Find the value of \(k\).