Eigensystem[{{6,-2},{-2,3}}]

Show[Plot3D[6 (x1^2)- 4 x1 x2+3 (x2^2),{x1,-2,2},{x2,-2,2},PlotStyle->{{Opacity[0.15]}}],ParametricPlot3D[{Cos[t],Sin[t],(6 (x1^2)- 4 x1 x2+3 (x2^2))/.{x1->Cos[t],x2->Sin[t]}},{t,0,2 Pi},PlotStyle->{{Thickness[0.01],Blue}}],Graphics3D[{Red,PointSize[0.05],Point[{((-2)/(Sqrt[5])),(1/(Sqrt[5])),7}]}],BoxRatios->{1,1,2},PlotRange->{-1,10}]