"Sample code: Using ParametrixPlot[], Point[] and Show[] to display the locus / trajectories of an ellipse, its center located at C(h,k) and its foci ";
h=2;k=3;
T=1;a=1;b=0.5;omega=2Pi/T;

(*** define focal points*)
f1=Sqrt[a^2-b^2];f2=-f1;
f1=f1+h;f2=h+f2;
F1={f1,k};F2={f2,k};
(*** end of define focal points*)

x[t_]:=a*Cos[omega*t]+h;y[t_]:=b*Sin[omega*t]+k;
prp1=ParametricPlot[{x[t],y[t]},{t,0,T}];
pp2=Graphics[{PointSize->0.03,Point[{h,k}]}];
ppf=Graphics[{PointSize->0.02,Point[{F1,F2}]}];
Show[{prp1,ppf,pp2}]