c[t_] := { (t^2), (t^3)}
d = c[u] - {x, y};
res = 
      Resultant[d[[1]], 
        d[[2]], u]

pp = 
      ParametricPlot[c[t], {t, -1, 1}, 
        PlotRange -> {{-1, 1}, {-1, 
              1}}]
pc = 
      ContourPlot[res, {x, -1, 1}, {y, -1, 1}, 
        Contours -> {0}]
pd = 
      ContourPlot[res, {x, -1, 1}, {y, -1, 1}, 
        PlotPoints -> 60]
Show[pd, pp, DisplayFunction -> $DisplayFunction]

num[t_] := {2 t, 1 - (t^2)}
den[t_] := (1 + 1  (t^2))
c[t_] := num[t]/den[t]
d = num[u] - {x, y} den[u];
res = 
      Resultant[d[[1]], 
        d[[2]], u]

pp = 
      ParametricPlot[c[t], {t, -1, 1}, 
        PlotRange -> {{-1, 1}, {-1, 
              1}}]
pc = 
      ContourPlot[res, {x, -1, 1}, {y, -1, 1}, 
        Contours -> {0}]
pd = 
      ContourPlot[res, {x, -1, 1}, {y, -1, 1}, 
        PlotPoints -> 60]
Show[pd, pp, DisplayFunction -> $DisplayFunction]