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 $|4x+3y|\leq 1000$?
$\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$
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 $(-7,1)$, which is 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 C-273)