Difference between revisions of "2009 AMC 12A Problems/Problem 16"
(New page: == Problem == A circle with center <math>C</math> is tangent to the positive <math>x</math> and <math>y</math>-axes and externally tangent to the circle centered at <math>(3,0)</math> with...) |
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Simplifying, we obtain <math>r^2 - 8r + 8 = 0</math>. By [[Vieta's formulas]] the sum of the two roots of this equation is <math>\boxed{8}</math>. | Simplifying, we obtain <math>r^2 - 8r + 8 = 0</math>. By [[Vieta's formulas]] the sum of the two roots of this equation is <math>\boxed{8}</math>. | ||
− | (We should actually solve for <math>r</math> to verify that there are two distinct positive roots. In this case we get <math>r=4\pm 2\sqrt 2</math>.) | + | (We should actually solve for <math>r</math> to verify that there are two distinct positive roots. In this case we get <math>r=4\pm 2\sqrt 2</math>. 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.) |
<asy> | <asy> |
Revision as of 13:59, 5 August 2012
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 |
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 |