f1 = x^2 + a x y + y^2

f2 = x y + y^2 - x^2

Manipulate[
Plot3D[Evaluate[{f1,f2}/.a->AA],{x,-1,1},{y,-1,1},PlotStyle-> {Blue, Green}],
{AA,-10,10}]

sols =Solve[f1==f2,x]

params = {(1/2) (y-a y),y, f1/.x->(1/2) (y-a y)}

params2 = {0,y, f1/.x->0}

curve = ParametricPlot3D[params/.a->3,{y,-1,1},PlotStyle-> {Thick,Red}]

curve2 = ParametricPlot3D[params2/.a->3,{y,-1,1},PlotStyle-> {Thick,Red}]

p1 =Plot3D[Evaluate[{f1,f2}/.a->3],{x,-1,1},{y,-1,1},PlotStyle-> {Blue, Green}]

Show[p1,curve,curve2]