A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda m g\) has one end attached to a fixed point \(O\) on a smooth horizontal surface. When a particle of mass \(m\) is attached to the free end of the string, it moves with speed \(v\) in a horizontal circle with centre \(O\) and radius \(x\). When, instead, a particle of mass \(2 m\) is attached to the free end of the string, this particle moves with speed \(\frac{1}{2} v\) in a horizontal circle with centre \(O\) and radius \(\frac{3}{4} x\).
(a) Find \(x\) in terms of \(a\).

(b) Given that \(v=\sqrt{12 a g}\), find the value of \(\lambda\).