1983 AHSME Problems/Problem 22
Problem
Consider the two functions and , where the variable and the constants and are real numbers. Each such pair of constants and may be considered as a point in an -plane. Let be the set of such points for which the graphs of and do not intersect (in the -plane). The area of is
Solution
We must describe geometrically those for which the equation , i.e. , has no solutions (equivalent to the graphs not intersecting). By considering the discriminant of this quadratic equation, there are no solutions if and only if . Thus is the unit circle (without its boundary, due to the inequality sign being rather than , but this makes no difference to the area), whose area is , so the answer is .