$x = a\sin \omega t$ or ${x \over a} = \sin \omega t$    ....(1)
$y = a\cos \omega t$ or ${y \over a} = \cos \omega t$    ....(2)

Squaring and adding, we get
${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{a^2}}} = 1$    ($\therefore$ ${\cos ^2}\omega t + {\sin ^2}\omega t = 1$)
or ${x^2} + {y^2} = {a^2}$

This is the equation of a circle. Hence particle follows a circular path.