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. You can also tell that there are two positive roots based on the visual interpretation.)
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 |
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