2014 AMC 12A Problem 25

Since $(0,0)$ is the midpoint of $A=(4,3)$ and $B=(-4,-3),$ the segment $AB$ is the latus rectum, so the directrix is parallel to $AB$ at distance $5$ on the far side, namely $4y-3x+25=0.$ Equating distances to focus and directrix gives $(4x+3y)^2$ $=25\big(25+2(4y-3x)\big).$ Writing $4x+3y=5s$ forces $s$ to be a multiple of $5,$ and $s=5t$ forces $t$ odd; with $t=2u+1$ the integer points are $$\begin{aligned} x&=-6u^2+2u+4,\\ y&=8u^2+14u+3. \end{aligned}$$ Then $|4x+3y|=|50u+25|\le1000$ iff $|2u+1|\le39,$ i.e. $-20\le u\le19.$ That gives $40$ lattice points. Thus, the correct answer is B.