2004 AMC 12B Problems/Problem 18
Problem
Points and are on the parabola , and the origin is the midpoint of . What is the length of ?
Solution
Let the coordinates of be . As lies on the parabola, we have . As the origin is the midpoint of , the coordinates of are . We need to choose so that will lie on the parabola as well. In other words, we need .
Substituting for , we get: .
This simplifies to , which solves to . Both roots lead to the same pair of points: and . Their distance is .
Alternate Solution
Let the coordinates of and be and , respectively. Since the median of the points lies on the origin, and expanding , we find:
It also follows that . Expanding this, we find:
To find the distance between the points, must be found. Expanding : we find the distance to be . Expanding this yields .
See Also
2004 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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All AMC 12 Problems and Solutions |
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