2004 AMC 12B Problems/Problem 21
Problem
The graph of is an ellipse in the first quadrant of the -plane. Let and be the maximum and minimum values of over all points on the ellipse. What is the value of ?
Solution
represents the slope of a line passing through the origin. It follows that since a line intersects the ellipse at either or points, the minimum and maximum are given when the line is a tangent, with only one point of intersection. Substituting, Rearranging by the degree of , Since the line is tangent to the ellipse, we want the discriminant, to be equal to . We want , which is the sum of the roots of the above quadratic. By Vieta’s formulas, that is .
See also
2004 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 |
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