Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers Class 9

Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers Class 9

Question 1.

Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers x, y, z must be simultaneously zero. Class 9

Solution:

Given, x + y + z = 0

Squaring on both sides, we get

(x + y + z)² = 0

∴ x² + y² + z² + 2(xy + yz + zx) = 0

∴ x² + y² + z² + 2(0) = 0 ...[xy + yz + zx = 0, ...(given)]

∴ x² + y² + z² = 0

Now, x², y², z² are all non-negative.

Sum of three non-negative numbers is zero only when each is zero.

∴ x² = 0, y² = 0, z² = 0

∴ x = 0, y = 0, z = 0

Hence, all three rational numbers must be simultaneously zero.