Solve for trajectory again

We have placed a particle of mass $$m$$ at $$\overset { \rightarrow }{ r } ={ r }_{ 0 }\hat { i }$$.

At $$t=0$$ we have given it a velocity of $$\overrightarrow { v } ={ v }_{ 0 }\hat { j }$$.In the whole co-ordinate plane the force is given by $$\overrightarrow{F}=-\overrightarrow{r}Newton$$

The title says solve for trajectory but I am telling you that the trajectory of the particle is ellipse.If the eccentricity of the ellipse is given by $$e=\frac {\sqrt {a} }{ b }$$ where $$a,b$$ are co-prime integers and a is square free then find $$a\times b$$

Details and Assumptions

1)$${ v }_{ 0 }=2 m/s,{ r }_{ 0 }=1 metre,m=1Kg$$

2)There are no other force acting on the particle except the force given.

3) Ignore gravity

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