A bead of mass \(m\) moves on a smooth circular wire, with centre \(O\) and radius \(a\), in a vertical plane. The bead has speed \(v_A\) when it is at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the downward vertical through \(O\), and \(\cos\alpha=\dfrac35\).

Subsequently, the bead has speed \(v_B\) at the point \(B\), where \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). Angle \(AOB\) is a right angle.

The reaction of the wire on the bead at \(B\) is in the direction \(OB\) and has magnitude equal to \(\dfrac16\) of the magnitude of the reaction when the bead is at \(A\).

(a) Find, in terms of \(m\) and \(g\), the magnitude of the reaction at \(B\).

(b) Given that \(v_A=\sqrt{kag}\), find the value of \(k\).