The resultant-displacement can be find by adding $x$ and $y$-components.
According to variation of $x$ and $y$, trajectory will be predicted, so resultant displacement is $y^{\prime}=(x+y)$
As given that,
$$
\begin{aligned}
&x=a \cos \omega t \\
&y=a \sin \omega t
\end{aligned}
$$
So, $y^{\prime}=(a \cos \omega t+a \sin \omega t)$
$$
\begin{aligned}
&\Rightarrow y^{\prime}=a(\cos \omega t+\sin \omega t) \\
&y^{\prime}=a \sqrt{2}\left[\frac{\cos \omega t}{\sqrt{2}}+\frac{\sin \omega t}{\sqrt{2}}\right] \\
&y^{\prime}=a \sqrt{2}\left(\cos \omega t \cos 45^{\circ}+\sin \omega t \sin 45^{\circ}\right) \\
&\quad=a \sqrt{2} \cos \left(\omega t-45^{\circ}\right)
\end{aligned}
$$
So, the displacement is not straight line and not parabola also.
Now, squaring and adding eqs. (i) and (ii),
$$
x^2+y^2=a^2 \quad\left[\because \cos ^2 \omega t+\sin ^2 \omega t=1\right]
$$
This is the equation of a circle, so motion in circular (independent of time).
Clearly, the locus is a circle of constant radius $a$.