2014 AMC 12A Problems/Problem 25

Problem

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$ . For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\leq 1000$ ?

$\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

Solution

The parabola is symmetric through $y=- \frac{4}{3}x$ , and the common distance is $5$ , so the directrix is the line through $(1,7)$ and $(-7,1)$ . That's the line $3x-4y = -25.$ Using the point-line distance formula, the parabola is the locus $x^2+y^2 = \frac{\left\lvert 3x-4y+25 \right\rvert^2}{3^2+4^2}$ which rearranges to $(4x+3y)^2 = 25(6x-8y+25)$ .

Let $m = 4x+3y \in \mathbb Z$ , $\left\lvert m \right\rvert \le 1000$ . Put $m = 25k$ to obtain $25k^2 = 6x-8y+25$ $25k = 4x+3y.$ and accordingly we find by solving the system that $x = \frac{1}{2} (3k^2-3) + 4k$ and $y = -2k^2+3k+2$ .

One can show that the values of $k$ that make $(x,y)$ an integer pair are precisely odd integers $k$ . For $\left\lvert 25k \right\rvert \le 1000$ this is $k= -39,-37,-35,\dots,39$ , so $40$ values work and the answer is $\boxed{\textbf{(B)}}$ .

(Solution by v_Enhance)

Solution 2

Consider the rotation of axes such that the axes are the lines passing through the origin with slope $\dfrac{3}{4}$ and $-\dfrac{4}{3}$ for x-axis and y-axis, respectively, and let the point on the rotated axis be $(x_1, y_1)$ . We can check that $x=\dfrac{4}{5}x_1-\dfrac{3}{5}y_1$ and $y=\dfrac{3}{5}x_1+\dfrac{4}{5}y_1$ 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 $ax_0+by_0+c=0$ and $(x_0, y_0)$ is the point. We have the focus as $(0,0)$ and $(5,0)$ and $(-5,0)$ as points on the parabola(on the rotated axes). Therefore, the directrix is $y=\pm 5$ , and it doesn't matter which one(due to the absolute value) so WLOG we choose $y_1=-5$ . The vertex is the midpoint between the focus and the foot of the altitude from focus to directrix, so the vertex is $(0, -\dfrac{5}{2})$ . Therefore, the equation is $y_1=\dfrac{x_1^{2}}{10}-\dfrac{5}{2}$ , and from the equations above we have $|3x+4y|=5x_1$ , so $|{x_1}|<200$ . One can check with $4x+3y$ and $4y-3x$ that the only time $x$ and $y$ can both be integers is when $x_1$ and $y_1$ are both integer multiples of $\dfrac{1}{5}$ . Therefore, the only time is when $x_1$ is an odd multiple of 5, and this is obviously sufficient because $y_1$ is also a multiple of $5$ . The values that satisfy thus are $x={-195, -185, -175, ..., 195}$ , and there are $\boxed{(B) 40}$ such numbers.

(Solution by Shaddoll)

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2014 AMC 12A Problems/Problem 25

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Problem

Solution

Solution 2

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