Problem:
The graph of partitions the plane into several regions. What is the area of the bounded region?
Solution:
For nonnegative values of , the equation can be written , or
For nonnegative , the graph thus consists of two rays. For nonpositive values of , the equation can be written , or
This gives two more rays. The bounded region is therefore enclosed by a parallelogram whose height is , whose base is , and whose area is .
Note: Because satisfies the given equation if and only if does, the graph is symmetric with respect to the origin. The requested area is thus twice the area of the triangle enclosed by the -axis and the two rays found for nonnegative .
The problems on this page are the property of the MAA's American Mathematics Competitions