AMC12 2021 A

AMC12 2021 A · Q2

AMC12 2021 A · Q2. It mainly tests Absolute value, Inequalities (AM-GM etc. basic).

Under what conditions is $\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?

对于实数$a$和$b$，在什么条件下$\sqrt{a^2+b^2}=a+b$成立？

(A) It is never true. It is never true.

(B) It is true if and only if $ab=0$. It is true if and only if $ab=0$.

(D) It is true if and only if $ab=0$ and $a+b\ge 0$. It is true if and only if $ab=0$ and $a+b\ge 0$.

(E) It is always true. It is always true.

Answer

Correct choice: (D)

正确答案：（D）

Solution

One can square both sides to get $a^{2}+b^{2}=a^{2}+2ab+b^{2}$. Then, $0=2ab\implies ab=0$. Also, it is clear that both sides of the equation must be nonnegative. The answer is $\boxed{\textbf{(D)}}$.

两边平方，得$a^{2}+b^{2}=a^{2}+2ab+b^{2}$。则，$0=2ab\implies ab=0$。此外，方程两边显然必须非负。答案是$\boxed{\textbf{(D)}}$。

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