2014 AMC 12A Problem 25

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

(0,0)

is the midpoint of

(4,3)

and

(-4,-3),

so this chord is the latus rectum and the directrix is parallel to it

The directrix is

4y-3x+25=0

; parametrize the lattice points and reduce

|4x+3y|

to

|50u+25|\le1000

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.

Problem 25 in other years