(*Generate n random ellipses*)
n=4;L=3;
As={};bs={};cs={};ellipses={};
For[i=1,i<=n,i++,
P=Table[Table[RandomReal[{-L,L}],{2}],{2}];
A=Transpose[P].P+RandomReal[2]IdentityMatrix[2];
b=Table[RandomReal[{-3L,3L}],{2}];
c=RandomReal[{-4L,4L}];
ellipse={x,y}.A.{x,y}+2b.{x,y}+c<=0;
AppendTo[As,A];AppendTo[bs,b];AppendTo[cs,c];
AppendTo[ellipses,ellipse];
];
ellipsesPlot=Show[Table[RegionPlot[ellipses[[i]],{x,-3L,3L},{y,-3L,3L},PlotPoints->100,PlotStyle->{Opacity[.4]}],{i,1,n}]]

(*SDP conditions*)
tau=Array[t,n];
taunonneg=And@@Table[tau[[i]]>=0,{i,1,n}];
cond1=And@@Table[tau[[i]]*ArrayFlatten[{{As[[i]],Transpose[{bs[[i]]}]},{{bs[[i]]},cs[[i]]}}]-ArrayFlatten[{{IdentityMatrix[2],Transpose[-{{xc,yc}}]},{-{{xc,yc}},g}}](Underscript[\[VectorGreaterEqual], (SemidefiniteConeList[3])]) 0,{i,1,n}];
cond2=ArrayFlatten[{{IdentityMatrix[2],Transpose[{{xc,yc}}]},{{{xc,yc}},s+g}}](Underscript[\[VectorGreaterEqual], (SemidefiniteConeList[3])]) 0;
(*Solve SDP*)
sol=SemidefiniteOptimization[s,cond1&&cond2&&taunonneg,{s,xc,yc,t[1],t[2],t[3],t[4],g}]
(*Define and plot minimal circle containing all ellipses*)
mincirc={x,y}.{x,y}-2{xc,yc}.{x,y}+g<=0/.sol;
mincircPlot=RegionPlot[mincirc,{x,-3L,3L},{y,-3L,3L},PlotStyle->LightGray,BoundaryStyle->{Thick, Gray}];
center=ListPlot[{{xc,yc}}/.sol,PlotStyle->Red];
Show[mincircPlot,ellipsesPlot,center]