Solve for trajectory again

We have placed a particle of mass mm at r=r0i^\overset { \rightarrow }{ r } ={ r }_{ 0 }\hat { i }.

At t=0t=0 we have given it a velocity of v=v0j^\overrightarrow { v } ={ v }_{ 0 }\hat { j }.In the whole co-ordinate plane the force is given by F=rNewton\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=abe=\frac {\sqrt {a} }{ b } where a,ba,b are co-prime integers and a is square free then find a×ba\times b

Details and Assumptions

1)v0=2m/s,r0=1metre,m=1Kg{ 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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