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 $(a,b)$
• B. There are infinitely many solution pairs$(a,b)$
• C. There are exactly two solution pairs
• D. There is exactly one solution pair $(a,b)$

Answer 1

Answer: (B)

There are infinitely many solution pairs$(a,b)$

Given: $(1+a^2+b^2{)}^2=3(1+a^2+b^2)$

$\Rightarrow (1+a+b)=\pm \sqrt{3(1+a^2+b^2)}$ $------\left(1\right)$

As we know for any real a,b: $a^2+b^2\ge 0$

$\Rightarrow a^2+b^2+1\ge 1$

Therefore both sides are real in $equation$ $1$ and hence there are infinitely possible pairs of $(a,b).$