2021 AMC 12A Spring 第 2 题

The left side is a square root, so it can never be negative; this forces $a + b \ge 0$

Squaring gives $a^2 + b^2 = a^2 + 2ab + b^2,$ so $2ab = 0$

Because $\sqrt{a^2+b^2}$ is never negative, equality requires $a + b \ge 0.$ Squaring both sides gives $a^2 + b^2 = (a+b)^2$ $= a^2 + 2ab + b^2,$ which simplifies to $2ab = 0,$ i.e. $ab = 0.$

Conversely, if $ab = 0$ then $a^2 + b^2 = (a+b)^2,$ and if additionally $a + b \ge 0$ then $\sqrt{a^2+b^2} = |a+b| = a+b.$ So both conditions together are exactly what is needed.

Thus, the correct answer is D.