Difference between revisions of "2023 AMC 12B Problems/Problem 10"
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Revision as of 02:09, 16 November 2023
Contents
Problem
In the -plane, a circle of radius with center on the positive -axis is tangent to the -axis at the origin, and a circle with radius with center on the positive -axis is tangent to the -axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
Solution 1
The center of the first circle is . The center of the second circle is . Thus, the slope of the line that passes through these two centers is .
Because this line is the perpendicular bisector of the line that passes through two intersecting points of two circles, the slope of the latter line is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2 (Coordinate Geometry)
The first circle can be written as we'll call this equation The second can we writen as , we'll call this equation
Expanding : Exapnding
Now we can set the equations equal to eachother: This is in slope intercept form therefore the slope is .
Video Solution 1 by OmegaLearn
See Also
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.