Question

If x+y+z=0 find the ratio of [x^4+y^4+z^4]/[x^2y^2+y^2z^2+z^2x^2]

Answer 1

x + y + z =0

then,
x² + y² + z² = 2(xy + yz + zx )

take square both sides

(x² + y² + z² )² =4(xy + yz +zx)²

x⁴ + y⁴ + z⁴ +2(x²y² + y²z² +z²x²) =4{x²y²+y²z² + z²x² +2(xy²z+yz²x+zx²y)}

x⁴ + y⁴ + z⁴ =2(x²y² +y²z² +z²x² ) +8xyz(x+y+x)

we know , according to question ,
x + y + z =0

(x⁴ +y⁴ +z⁴)/(x²y²+y²z² +z²x²) =2 + 8×0

(x⁴ + y⁴ +z⁴)/(x²y² +y²z² +z²x²) =2