Question

If x + 2y + 3z = 0 and x³ + 4y³ + 9z³ = 18xyz ; evaluate : 
`[( x + 2y )^2]/(xy) + [(2y + 3z)^2]/(yz) + [(3z + x)^2]/(zx)`

Answer 1

Given that x³ + 4y³ + 9z³ = 18xyz and x + 2y + 3z = 0
x + 2y = - 3z, 2y + 3z = -x and 3z + x = -2y
Now
`[( x + 2y )^2]/(xy) + [(2y + 3z)^2]/(yz) + [(3z + x)^2]/(zx)`

= `[(-3z)^2]/(xy) + [(-x)^2]/(yz) + (-2y)^2/(zx)`

= `(9z^2)/(xy) + (x^2)/(yz) + (4y^2)/(zx)`

= `[ x^3 + 4y^3 + 9z^3 ]/[xyz]`

Given that x³ + 4y³ + 9z³ = 18xyz

∴ `[( x + 2y )^2]/(xy) + [(2y + 3z)^2]/(yz) + [(3z + x)^2]/(zx)`

= `[18xyz]/[xyz]`

= 18