AMC 12 2013b Problem 21

Consider the set of $30$ parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?

\textbf{(A)}\ 720 \textbf{(B)}\ 760 \textbf{(C)}\ 810 \textbf{(D)}\ 840 \textbf{(E)}\ 870