Question

If 2x - y - z = 0. prove that 4x^2 - y^2 + z^2 =4xz​

Answer 1

Step-by-step explanation:

(x+y+z)²=x²+y²+z²+2xy+2yz+2zx

So x²+y²+z² = -2(xy+yz+zx)

So (x²+y²+z²)²=4(x²y²+y²z²+z²x²+2xyz(x+y+z)) .. {x+y+z=0)

(x²+y²+z²)²=x⁴+y⁴+z⁴ +2x²y² +2y²z²+2z²x²

So 4(x²y²+y²z²+z²x²)=x⁴+y⁴+z⁴+ 2(x²y²+y²z²+z²x²)

So x⁴+y⁴+z⁴ = 2(x²y²+y²z²+z²x²)

So (x²y²+y²z²+z²x²)/(x⁴+y⁴+z⁴)=(1/2)