2009 AMC 12A Problem 16

Below is the professionally curated solution for Problem 16 of the 2009 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

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Concepts:tangent circles coordinate geometry Vieta’s Formulas

Difficulty rating: 1910

16.

A circle with center $C$ is tangent to the positive $x$ - and $y$ -axes and externally tangent to the circle centered at $(3, 0)$ with radius $1.$ What is the sum of all possible radii of the circle with center $C?$

$3$

$4$

$6$

$8$

$9$

Solution:

A circle tangent to both positive axes with radius $r$ has center $(r, r).$ External tangency to the circle at $(3, 0)$ of radius $1$ means the distance between centers is $r + 1$ : $(r - 3)^2 + r^2 = (r + 1)^2.$

Expanding gives $r^2 - 8r + 8 = 0.$ Both roots $r = 4 \pm 2\sqrt{2}$ are positive, and by Vieta's formulas their sum is $8.$

Thus, the correct answer is D.

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