Displacement vector $\vec{r}=x \hat{i}+y \hat{j}$

here $\vec{r}=($ Acos $\omega t) \hat{i}+($ Bsin $\omega t) \hat{j}$

Comparing with $\vec{r}=x \hat{i}+y \hat{j}$

$\frac{x}{\mathrm{~A}}=\cos \omega t \quad \frac{y}{\mathrm{~B}}=\sin \omega t$

$\cos ^{2} \omega t+\sin ^{2} \omega t=1$

$\left(\frac{x}{\mathrm{~A}}\right)^{2}+\left(\frac{y}{\mathrm{~B}}\right)^{2}=1$

$\frac{x^{2}}{\mathrm{~A}^{2}}+\frac{y^{2}}{\mathrm{~B}^{2}}=1$

Which is equation of ellipse.

$\therefore$ Trajectory of particle is an ellipse.

$(b)$ Velocity of the particle,

$v =\frac{d r}{d t}=\hat{i} \frac{d}{d t}(\mathrm{~A} \cos \omega t)+\hat{j} \frac{d}{d t}(\mathrm{~B} \sin \omega t)$

$=\hat{i}[\mathrm{~A}(-\sin \omega t) \cdot \omega]+\hat{j}[\mathrm{~B}(\cos \omega t) \cdot \omega]$

$=-\hat{i} \mathrm{~A} \omega \sin \omega t+\hat{j} \mathrm{~B} \omega \cos \omega t$

Acceleration of the particle $(a)$,

$(a)=\frac{d v}{d t}$