2017 AMC 10A Problems/Problem 17
Problem
Distinct points , , , lie on the circle and have integer coordinates. The distances and are irrational numbers. What is the greatest possible value of the ratio ?
Solution
Because , , , and are integers there are only a few coordinates that actually satisfy the equation. The coordinates are and We want to maximize and minimize They also have to be the square root of something, because they are both irrational. The greatest value of happens when it and are almost directly across from each other and are in different quadrants. For example, the endpoints of the segment could be and because the two points are almost across from each other. The least value of is when the two endpoints are in the same quadrant and are very close to each other. This can occur when, for example, is and is They are in the same quadrant and no other point on the circle with integer coordinates is closer to the point than Using the distance formula, we get that is and that is
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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