Motion of a particle can be described in x-direction by x = 1 sin (ot, and y-direction by y = b cos cot. The particle is moving on

The motion of the particle is described by the parametric equations x = a sin ωt and y = b cos ωt. To determine the path, we eliminate the time parameter 't' by rearranging the equations as x/a = sin ωt and y/b = cos ωt. Squaring and adding these equations yields (x/a)² + (y/b)² = sin² ωt + cos² ωt. Since sin² θ + cos² θ = 1, the resulting equation is (x²/a²) + (y²/b²) = 1. This is the standard mathematical equation for an ellipse centered at the origin [1]. These curves are known as Lissajous figures, which result from the superposition of two mutually perpendicular simple harmonic motions [1]. While the path becomes a circle if the amplitudes 'a' and 'b' are equal, in the general case where 'a' and 'b' are distinct, the particle follows an elliptical path [1].

Sources

1. [1] Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.) > Chapter 2: The Solar System > Kepler's Laws of Planetary Motion > p. 21