2009 AMC 12A Problems/Problem 16
Problem
A circle with center is tangent to the positive
and
-axes and externally tangent to the circle centered at
with radius
. What is the sum of all possible radii of the circle with center
?
Solution
Let be the radius of our circle. For it to be tangent to the positive
and
axes, we must have
. For the circle to be externally tangent to the circle centered at
with radius
, the distance between
and
must be exactly
.
By the Pythagorean theorem the distance between and
is
, hence we get the equation
.
Simplifying, we obtain . By Vieta's formulas the sum of the two roots of this equation is
.
(We should actually solve for to verify that there are two distinct positive roots. In this case we get
. This is generally a good rule of thumb, but is not necessary as all of the available answers are integers, and the equation obviously doesn't factor as integers.)
See Also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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All AMC 12 Problems and Solutions |