2014 AMC 12A Problems/Problem 25
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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 \[
\begin{align*}
25k^2 &= 6x-8y+25 \\
25k &= 4x+3y.
\end{align*}
\] 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)