A1 (2007)

gamma[t_]={Cos[t],Sin[2 t]};
ParametricPlot[gamma[t],{t,0,2 Pi-0.1},PlotStyle->Black]

c[t_]={Cos[t],Sin[t]};
ParametricPlot[c[t],{t,0,2 Pi},PlotStyle->Black]

Show[ParametricPlot[c[t],{t,0,2 Pi},PlotStyle->Black],ParametricPlot[{1,0}+(1/2) {Cos[t],Sin[t]},{t,0,2 Pi},PlotStyle->{Black,Dashed}],ParametricPlot[{1,0}+t{1/Sqrt[2],1/Sqrt[2]},{t,0,1/2},PlotStyle->Black], PlotRange->All]

ParametricPlot[{t^2,t^3},{t,-1/2^(1/3),1/2^(1/3)},PlotStyle->Black,AspectRatio->Automatic]

parabplot[alpha_]:=Show[ParametricPlot[{t,alpha t^2 + alpha t + 1/24},{t,-1/2-1,-1/2+1},PlotStyle->Black],ParametricPlot[{alpha t^2 + alpha t + 1/24,t},{t,-1/2-1,-1/2+1},PlotStyle->Black],PlotRange->{{-1/2-1, -1/2+1},{-1/2-1, -1/2+1}}]

parabplot[1]

Animate[parabplot[alpha],{alpha,-3,4}]

parabplot[(13-Sqrt[601])/12]

parabplot[(13+Sqrt[601])/12]

parabplot[2/3]

parabplot[3/2]