2014 AMC 12A Problems/Problem 25
Problem
The parabola has focus
and goes through the points
and
. For how many points
with integer coefficients is it true that
?
Solution
The parabola is symmetric through , and the common distance is
, so the directrix is the line through
and
. That's the line
Using the point-line distance formula, the parabola is the locus
which rearranges to
.
Let ,
. Put
to obtain
\[25k^2 &= 6x-8y+25\] (Error compiling LaTeX. Unknown error_msg)
\[25k &= 4x+3y.\] (Error compiling LaTeX. Unknown error_msg)
and accordingly we find by solving the system that and
.
One can show that the values of that make
an integer pair are precisely odd integers
. For
this is
, so
values work and the answer is
.
(Solution by v_Enhance)
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Question |
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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