Question

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 $|4x+3y|\le1000?$

$38$

$40$

$42$

$44$

$46$

Answer: B

Concepts:parabola lattice point Diophantine Equation

Difficulty rating: 2650

Solution:

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 $5\mid s,$ 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.

2014 AMC 12A Problem 25

Answer: B

Difficulty rating: 2650

Solution:

Problem 25 in Other Years