a1=3;b1=1;a2=2;b2=0.5;prange={{-6,6},{-6,6}};Manipulate[cen2={cX,cY};M=RotationMatrix[phi];e1=ParametricPlot[{a1*Cos[u],b1*Sin[u]},{u,0,2*Pi},PlotStyleBlack,PlotRangeprange];e2=ParametricPlot[cen2+M.{a2*Cos[v],b2*Sin[v]},{v,0,2*Pi},PlotStyleBlue,PlotRangeprange];minres=NMinimize[{Norm[{a1*Cos[u],b1*Sin[u]}-(cen2+M.{a2*Cos[v],b2*Sin[v]})],0≤u≤2*Pi&&0≤v≤2*Pi},{u,v},Method"RandomSearch"];p1={a1*Cos[u],b1*Sin[u]}/.minres[[2]];p2=cen2+M.{a2*Cos[v],b2*Sin[v]}/.minres[[2]];lin=Graphics[{Red,Thickness[0.005],Line[{p1,p2}]}];Show[e1,e2,lin,AspectRatio1],{cX,-5,5},{cY,-5,5},{phi,0,2*Pi},TrackedSymbols{cX,cY,phi}]