A particle of mass m oscillates along x -axis according to equation...

According to the question x=\operatorname{arsin} $\omega$t $\rightarrow \frac{x}{a}=\sin \omega$t Now velocity $\mathrm{V}=\frac{d x}{d t}=a \omega \cos \omega t\Rightarrow \quad \frac{\mathrm{V}}{a \omega}=\cos \omega t$ From eq. (i) and (ii) \begin{gathered}\frac{x^2}{a^2}+\frac{v^2}{a^2 \omega^2}=\sin ^2 \omegat+\cos ^2 \omegat \\ \frac{x^2}{a^2}+\frac{v^2}{a^2 \omega^2}=1 \end{gathered} \Rightarrow This is the equation of ellipse. So the graph between moment and displacement of particle is ellipse.