If $2 (a^{2} + b^{2}) = (a + b)^{2}$ , then which of the following is true?

a = b

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a = - b

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a = 2 b + 3

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None of the above

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Solution

The correct option is A $a = b$
Given, $2 a^{2} + 2 b^{2} = a^{2} + 2 a b + b^{2}$ .

We know, $(a + b)^{2} = a^{2} + 2 a b + b^{2}$
and $(a - b)^{2} = a^{2} - 2 a b + b^{2}$ .

The value of $2 (a^{2} + b^{2}) = (a + b)^{2}$ is equal to:
$2 a^{2} + 2 b^{2} = a^{2} + 2 a b + b^{2}$
$\Rightarrow 2 a^{2} + 2 b^{2} - a^{2} - 2 a b - b^{2} = 0$
$\Rightarrow a^{2} + b^{2} - 2 a b = 0$
$\Rightarrow (a - b)^{2} = 0$
$\Rightarrow a + b = 0$
$\Rightarrow a = b$ .

Therefore, option $A$ is correct.

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