2004 AMC 12B Problems/Problem 21
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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 
, 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 • 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 | ||
