At any instant the velocity of a particle of mass $500\,g$ is $\left(2 t \hat{ i }+3 t ^2 \hat{ j }\right)\,ms ^{-1}$. If the force acting on the particle at $t=1 s$ is $(\hat{i}+x \hat{j}) N$. Then the value of $x$ will be:

The displacement vector of a particle of mass $m$ is given by $\vec r (t)\, = \,A\,\cos \,\omega t\,\hat i\, + \,B\,\sin \,\omega t\,\hat j$.

$(a)$ Show that the trajectory is an ellipse.

$(b)$ Show that $\vec F =-\,m\omega ^2\vec r $.