Construction of Hyperbola

y[t_, d_] :=
  Module[{i = -16 + 8 d^2 - d^4 + 16 d t - 4 d^3 t + 16 t^2 -
      4 d^2 t^2}, If[i >= 0, 1/4  Sqrt[i], 10^3]];
x[t_, d_] := 1/4 (-d^2 - 2 d t);

Manipulate[Show[Graphics[{Darker@Red,
    Point[{x[t, d], y[t, d]}], Point[{x[t, d], -y[t, d]}],
    Darker@Blue,
    Circle[{1, 0}, Max[0, t + d]],
    Darker@Purple,
    Circle[{-1, 0}, Max[0, t]]
    }, PlotRange -> {{-2, 2}, {-2, 2}}],
  If[t >= (2 - d)/2,
   {ParametricPlot[{x[tt, d], y[tt, d]}, {tt, (2 - d)/2, t}],
    ParametricPlot[{x[tt, d], -y[tt, d]}, {tt, (2 - d)/2, t}]},
   Graphics[]
   ]
  ], {t, -1.2, 12}, {{d, 1.2}, 0, 2}]