Problem:
The parabola P has focus (0,0) and goes through the points (4,3) and (−4,−3). For how many points (x,y)∈P with integer coordinates is it true that ∣4x+3y∣≤1000?
Answer Choices:
A. 38
B. 40
C. 42
D. 44
E. 46 Solution:
Let O=(0,0),A=(4,3), and B=(−4,−3). Because A,B∈P and O is the midpoint of AB, it follows that AB is the latus rectum of the parabola P. Thus the directrix is parallel to AB. Let T be the foot of the perpendicular from O to the directrix of P. Because OT=OA=OB=5 and OT is perpendicular to AB, it follows that T=(3,−4). Thus the equation of the directrix is y+4=43(x−3), and in general form the equation is 4y−3x+25=0.
Using the formula for the distance from a point to a line, as well as the definition of P as the locus of points equidistant from O and the directrix, the equation of P is
x2+y2=42+32∣4y−3x+25∣
After squaring and rearranging, this is equivalent to
Assume x and y are integers. Then 4x+3y is divisible by 5 . If 4x+3y=5s for s∈Z, then 2s2=50+16y−12x=50+16y−3(5s−3y)=50+25y−15s. Thus s is divisible by 5 . If s=5t for t∈Z, then 2t2=2+y−3t, and so y=2t2+3t−2. In addition 4x=5s−3y=25t−3y=25t−3(2t2+3t−2)=−6t2+16t+6, and thus t is odd. If t=2u+1 for u∈Z, then
x=−6u2+2u+4 and y=8u2+14u+3.
Conversely, if x and y are defined as in (2) for u∈Z, then x and y are integers and they satisfy (1), which is the equation of P. Lastly, with u∈Z,