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 $|4x+3y|\leq 1000$?
+336 1250n+251250n+25 to 1250n+12251250n+1225. It turns out that only when n=0n=0 or 11 are the criteria satisfied, and in each case there are 2020 of the pairs. So 40 is the answer.
For more in debt answer just ask meh
-wolfie
(If wrong im sorry my brain hurts )
And
congrats on getting a amc 12 problem correct, the hardest problem on the test was correct,y answered by you. You are surely underrated
+118673 Concave up parabola
Focus(0,0) A point (4,3)
Distance of the point to the focus is 5
Distance of (4,3) to the directrix is 5
So the directrix is y=-2
So the vertex is (0,-1)
Equation is
\((x-0)^2=4a(y+1)\\ x^2=4(y+1)\\ so\\ y=\frac{x^2}{4}-1=\frac{x^2-4}{4}\)
\(4x+3y \le1000\\ 4x+3(\frac{x^2-4}{4})\le1000\\ 16x+3x^2-12\le4000\\ 16x+3x^2-12\le4000\\ 3x^2+16x-4012\le 0\\ find\;\;roots\\ x=\frac{-16\pm\sqrt{256+12*4012}}{6}\\ x=\frac{-16\pm 220}{6}\\ x=-39.\dot3\;\;or\;\;\;x=34\)
So this will be true for all integer values of x from -39 to 34 = 34- -39+1 = 74 integer x values.
BUT how many of these integer x values will have integer y values.
Only those where x^2 is divisible by 4. If x^2 is divisible by 4 then x is divisible by 2. So x must be even.
So how many even numbers are there from -38 to 34 inclusive? 17+19+1 = 37
So 37 coordinate pairs will make this statement true.
Here is the graph.
