Question

Consider the equation $(1 + a + b)^2 =3(1 + a^2 + b^2)$ where $a, b$ are real numbers. Then,

• A. there is no solution pair (a, b)
• B. there are infinitely many solution pairs (a, b)
• C. there are exactly two solution pairs (a, b)
• D. there is exactly one solution pair (a, b)

Answer 1

Answer: (D)

Given,

$( 1 + a + b)^2 = 3( 1 + a^2 + b^2)$

$1 + a^2 + b^2 + 2a + 2b + 2ab$

$ = 3 + 3a^2 + 3b^2$

$\Rightarrow 2a^2 - 2b^2 - 2a - 2b - 2ab + 2 = 0$

$\Rightarrow (a^2 - 2a + 1) + (b^2 - 2b + 1)$

$+(a^2 + b^2 - 2ab)= 0$

$\Rightarrow (a - 1)^2 + (b - 1)^2 + ( a - b)^2 = 0$

$\therefore a - 1 = 0, b - 1 = 0, a - b = 0$

$\Rightarrow a = 1, b = 1, a = b$

$\therefore a = b = 1 $

Exactly one pair.