Difference between revisions of "1984 AHSME Problems/Problem 22"
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Latest revision as of 12:51, 5 July 2013
Problem
Let and
be fixed positive numbers. For each real number
let
be the vertex of the parabola
. If the set of the vertices
for all real numbers of
is graphed on the plane, the graph is
Solution
The x-coordinate of the vertex of a parabola is , so
. Plugging this into
yields
, so
. Notice that
, so all of the vertices are on a parabola. However, we have only showed that all of the points in the locus of vertices are on a parabola, we have not shown whether or not all points on the parabola are on the locus. Assume we are given an
on the parabola.
,
, so a unique
, and therefore a unique vertex, is determined for each point on the parabola. We therefore conclude that every point in the locus is on the parabola and every point on the parabola is in the locus, and the graph of the locus is the same as the graph of the parabola,
.
See Also
1984 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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