c
\(\frac{x}{a}=\sin \omega t\)

\(\Rightarrow \cos \omega t=\sqrt{1-\left(\frac{x_{2}}{a^{2}}\right)}\)

\(\frac{y}{a}=\sin 2 \omega t\)

\(=2 \sin \omega t \times \cos \omega t\)

\(=2 \frac{x}{a} \times \sqrt{1-\left(\frac{x^{2}}{a^{2}}\right)}\)

\( \Rightarrow y = \frac{{2x}}{{{a^2}}}\sqrt {\left( {a - x} \right)\left( {a + x} \right)} \)

Hence trajectory of particle will look like as \((\mathrm{c})\)