Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Find the answer

3. A line segment $AB$ of length 4 moves along the positive x-axis, and another line segment $CD$ of length 2 moves along the positive y-axis. If the four endpoints $A$ , $B$ , $C$ , and $D$ are concyclic, then the locus of the center of this circle is $\qquad$

Your answer

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

Solution

3. $x^{2}-y^{2}=3(x>2, y>1)$ .

As shown in Figure 2, let the center of the circle be $M(x, y)(x>2, y>1)$ . Then $A(x-2,0)$ ,
$B(x+2,0)$ .
$C(0, y-1)$ ,
$D(0, y+1)$ .
From $|M A|^{2}=|M C|^{2}$ ,
we get $2^{2}+y^{2}=x^{2}+1^{2}$ .
Therefore, the required locus is a segment of a hyperbola, whose equation is $x^{2}-y^{2}=3(x>2, y>1)$ .

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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.