Translating this Maple-like pseudo-code into
(* parametric plot (r*(1-t^2)/(1+t^2),r*2\[Times]t/(1+t^2)),t=-2..2,r=10 *)
With[{r = 10},
ParametricPlot[r/(1 + t^2) {(1 - t^2), 2 t}, {t, -90, 90}, PlotRange -> All]
]
you see it will not really close the circle at {-r,0} because of
In[48]:= r Limit[(1 - t^2)/(1 + t^2), t -> Infinity]
Out[48]= -r
Check your parametrization, you cannot - at least not with Mathematica - do a parametric plot with t reaching from -Infinity to +Infinity.
