Difference between revisions of "2004 AMC 12B Problems/Problem 18"
(→See Also) |
|||
Line 45: | Line 45: | ||
{{AMC12 box|year=2004|ab=B|num-b=17|num-a=19}} | {{AMC12 box|year=2004|ab=B|num-b=17|num-a=19}} | ||
+ | {{MAA Notice}} |
Revision as of 19:58, 3 July 2013
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 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.