2014 AMC 12A Problems/Problem 25
Contents
[hide]Problem
The parabola has focus and goes through the points and . For how many points with integer coordinates 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 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)
Solution 2
Consider the rotation of axes such that the axes are the lines passing through the origin with slope and for x-axis and y-axis, respectively, and let the point on the rotated axis be . We can check that and by the distance from a point to line formula $\dfrac{ax_0+by_0+c}{\sqrt{a^{2}+b^{2}}$ (Error compiling LaTeX. Unknown error_msg) where the equation of the line is and is the point. We have the focus as and and as points on the parabola(on the rotated axes). Therefore, the directrix is , and it doesn't matter which one(due to the absolute value) so WLOG we choose . The vertex is the midpoint between the focus and the foot of the altitude from focus to directrix, so the vertex is . Therefore, the equation is , and from the equations above we have , so . One can check with and that the only time and can both be integers is when and are both integer multiples of . Therefore, the only time is when is an odd multiple of 5, and this is obviously sufficient because is also a multiple of . The values that satisfy thus are , and there are such numbers.
(Solution by Shaddoll)
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
1) The line of symmetry is NOT y= -x but 4x + 3y = 0
2) In the expression for x, it is NOT 8 but 8k.
With these minor corrections, the solution still holds good.