2006 AMC 12A Problems/Problem 19
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Problem
Circles with centers and
have radii
and
, respectively. The equation of a common external tangent to the circles can be written in the form
with
. What is
?
![AMC12 2006A 19.png](https://wiki-images.artofproblemsolving.com//8/8d/AMC12_2006A_19.png)
Solution
Let be the line that goes through
and
, and let
be the line
. If we let
be the measure of the acute angle formed by
and the x-axis, then
.
clearly bisects the angle formed by
and the x-axis, so
. We also know that
and
intersect at a point on the x-axis. The equation of
is
, so the coordinate of this point is
. Hence the equation of
is
, so
, and our answer choice is
.
See also
2006 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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