LetF=(x2+y2+z2)+λ(xy+yz+zx)=0
Fx=0,Fy=0,Fz=0
2x+λ(y+z)=0,2y+λ(x+z)=0,2z+λ(x+y)=0
λ=y+z−2x,λ=x+z−2y,λ=x+y−2z
Equating the equations ,we have,
y+z−2x,=x+z−2y,=x+y−2z
Taking,
y+zx,=x+zy
We have, (x−y)(x+y+z)=0
x=yorx+y+z=0
we must have x=y since x+y+z cannot be 0
Suppose x+y+z=0, squaring this we have,
(x2+y2+z2)+2(xy+yz+zx)=0
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