Problem:
A circle has center and radius 13. Another circle has center and radius . The line passing through the two points of intersection of the two circles has equation . What is ?
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Solution:
The first circle has equation , and the second circle has equation . Expanding these two equations, subtracting, and simplifying yields . Because the points of intersection of the two circles must satisfy this new equation, it must be the required equation of the line through those points, so . In fact, the circles intersect at and .
The problems on this page are the property of the MAA's American Mathematics Competitions