Manipulate[Show[{ParametricPlot[
{Re[(z0[[1]]+I z0[[2]]+(Exp[r]-1)/(Exp[r]+1)Exp[I t])/(1+Conjugate[z0[[1]]+I z0[[2]]]  (Exp[r]-1)/(Exp[r]+1)Exp[I t])],
Im[(z0[[1]]+I z0[[2]]+(Exp[r]-1)/(Exp[r]+1) Exp[I t])/(1+Conjugate[z0[[1]]+I z0[[2]]]  (Exp[r]-1)/(Exp[r]+1)Exp[I t])]},
{t,0,2 Pi},Axes->True,PlotRange->{{-1,1},{-1,1}}],Graphics[Circle[{0,0},1]]}],{{r,1},0,5},{{z0,{0,0}},Locator},Deployed->True]