Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

11. If $a, b \in \mathbf{R}$ and $a^{2}+b^{2}=10$ , then the range of values for $a-b$ is $\qquad$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

11. $[-2 \sqrt{5}, 2 \sqrt{5}]$ .

Given $a, b \in \mathbf{R}$ and $a^{2}+b^{2}=10$ , we have
$\begin{array}{l} (a-b)^{2}=2\left(a^{2}+b^{2}\right)-(a+b)^{2} \\ \leqslant 2\left(a^{2}+b^{2}\right)=20 . \end{array}$

Thus, $|a-b| \leqslant 2 \sqrt{5}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.