2011 AMC 12B Problems/Problem 14
Contents
Problem
A segment through the focus of a parabola with vertex
is perpendicular to
and intersects the parabola in points
and
. What is
?
Solution 1
Name the directrix of the parabola . Define
to be the distance between a point
and a line
.
Now we remember the geometric definition of a parabola: given any line (called the directrix) and any point
(called the focus), the parabola corresponding to the given directrix and focus is the locus of the points that are equidistant from
and
. Therefore
. Let this distance be
. Now note that
, so
. Therefore
. We now use the Pythagorean Theorem on triangle
;
. Similarly,
. We now use the Law of Cosines:
This shows that the answer is .
Solution 2
WLOG we can assume that the parabola is . Therefore
and
. Also
and
.
and
by the pythagorean theorem.
Now using the law of cosines on we have:
and as desired.
solution by mihirb
See also
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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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