Problem:
A graph is symmetric about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers , where and and are not both 0 , is the graph of
symmetric about the line ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
There are two cases, depending on whether the denominator of the fraction is constant.
CASE 1. Suppose that . Then the graph of is a line. It is symmetric with respect to the line if the slope of the line is 1 and the line passes through the origin, or the slope is -1 . A line with slope 1 that passes through the origin occurs if and . There are 10 possible values of . The slope is -1 if . There are 10 possible values of , and 11 possible values of , which gives 110 possibilities. This gives a total of 120 possibilities in Case 1 .
CASE 2. Suppose that . The quantity tends to as . This is a horizontal asymptote. The degree of the polynomial in the numerator can be reduced by subtracting this asymptotic value:
where
If , then the graph of consists of the points , except that the point is missing because the formula reads when . In any case, this set of points is not symmetric with respect to the line . From now on, assume that . Multiply both sides by to find that
Note that because . Thus, if it were desired, both sides of the new equation could be divided by to recover the former equation. Hence the set of pairs that satisfy the former equation is exactly the same as the set of that satisfy the new equation, which determines a rectangular hyperbola. This hyperbola is symmetric with respect to the line if and only if its center lies on this line of symmetry. That is, . Thus if and only if .
If , then and are free to take on any nonzero values, which gives quadruples . If , then because and , there are quadruples to consider. Note that if and only if either and and have opposite signs, which is cases, or and or , which is additional cases. Thus the total number of possibilities in Case 2 is 1172.
In all there are quadruples satisfying the given conditions.
Note: Functions of the form with are known as "linear-fractional" or "MΓΆbius". They arise in complex analysis and in number theory.
The problems on this page are the property of the MAA's American Mathematics Competitions