Clear[x,y]

Eigensystem[{{-2,1},{0,2}}]

sol1=DSolve[{x'[t]==-2x[t]+y[t],y'[t]==2y[t],x[0]==1,y[0]==0.01},{x[t],y[t]},t];{x1[t_],y1[t_]}={x[t],y[t]}/.sol1[[1]];

sol2=DSolve[{x'[t]==-2x[t]+y[t],y'[t]==2y[t],x[0]==1,y[0]==-0.01},{x[t],y[t]},t];{x2[t_],y2[t_]}={x[t],y[t]}/.sol2[[1]];

Manipulate[ParametricPlot[{{x2[t],y2[t]},{x1[t],y1[t]}},{t,0,2},PlotRange->All,PlotLabel->N@Round[Norm[{x1[T],y1[T]}-{x2[T],y2[T]}],1/100],Epilog->{AbsolutePointSize[6],Point[{x1[T],y1[T]}],Point[{x2[T],y2[T]}]}],{T,0,2,.1}]

Solve[Simplify[Norm[{x1[t],y1[t]}-{x2[t],y2[t]}],t≥0]==1,t,Reals][[2,1,2]]