Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

Example 4 If $3 x-y-1=0$ , find
$z=\left|\sqrt{x^{2}+y^{2}-8 x-2 y+17}-\sqrt{x^{2}+y^{2}-8 y+16}\right|$

the maximum value.

Your answer

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

Solution

Solution: The symmetric point of point $B$ with respect to the line $3 x-y-1=0$ is $B_{1}(3,3)$ . Then, by the knowledge of plane geometry, we have
$z_{\text {max }}\left|A B_{1}\right|=\sqrt{5} .$

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