Question

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?

$720$

$760$

$810$

$840$

$870$

Solution:

Two parabolas with common focus $O$ meet in exactly $2$ points, except when their directrices are parallel and $O$ lies outside the strip between them, in which case they do not meet. The non-intersecting pairs have directrices of equal slope and $y$ -intercepts of the same sign. There are $5$ slopes, and for each, $2\binom{3}{2} = 6$ same-sign intercept pairs. Since every intersecting pair meets in $2$ points and no point lies on three parabolas, the total is $2\left(\binom{30}{2} - 5\cdot 6\right) = 2(435 - 30) = 810.$ Thus, the correct answer is C.

2013 AMC 12B Problem 21

Solution:

Problem 21 in Other Years