Question

The magnitude of displacement of a particle moving in a circle of radius a with constant angular speed w varies with time t as....

Answer 1

see diagram

Let us say that the particle is at the left extreme point of the diameter (horizontal). call this point origin O.
The displacement vector is the line joining this point O and the current position P (x,y) or (r, theta)  of the particle on the circle.

  X  = a + a cos Ф = 2a cos² Ф/2
  S = √(2 a X) 

  S =    2 a  cos (wt / 2)

Position vector r or S is given by : r = 2a cos Ф/2 
     where Ф is the angle made by line joining P and center of circle C, with the
     horizontal.

Ф = w t