Question

If ` (x+y)/(p^(3)-q^(3)) = (y+z)/(q^(3)-r^(3)) = (z+x)/(r^(3)-p^(3))`, then prove that x + y + z = 0.

Answer 1

Let ` (x+y)/(p^(3)-q^(3)) = (y+z)/(q^(3)-r^(3)) = (z+x)/(r^(3)-p^(3)) = k`
` rArr x + y = k(p^(3)-q^(3)), y + z = k (q^(3)-r^(3)), z + x = k(r^(3)-p^(3))`
` rArr x + y + y + z + z +x = k (p^(3)-q^(3)+q^(3)-r^(3)+r^(3)-p^(3))`
` rArr 2 (x+y+z) = k (0)`
`:. x + y + x = 0`.