2014 AMC 12A Problem 25

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?

\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.