2021 AMC 12A Spring Problem 2

Question

Under what conditions is

\sqrt{a^2 + b^2} = a + b

true, where

a

and

b

are real numbers?

It is never true.

It is true if and only if

ab = 0.

It is true if and only if

a + b \ge 0.

It is true if and only if

ab = 0

and

a + b \ge 0.

It is always true.

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.

Problem 2 in other years