The velocity of a particle moving in the x-y plane is given by dx/dt = 8πsin(2πt) and dy/dt = 5πcos(2πt) where, t = 0, x = 8 and y = 0. The path of the particle is:

dx/dt = 8πsin(2πt) ⇒ x = -4cos(2πt) + C
Since at t=0, x=8, we have 8 = -4cos(0) + C, so C = 12. Therefore, x = -4cos(2πt) + 12.
Also, dy/dt = 5πcos(2πt) ⇒ y = 2.5sin(2πt) + D
Since at t=0, y=0, we have 0 = 2.5sin(0) + D, so D = 0. Therefore, y = 2.5sin(2πt).
Hence ((x-12)/4)² + (y/2.5)² = 1 which is the equation of an ellipse.