Problem:
What is the area of the region enclosed by the graph of the equation ?
Answer Choices:
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Solution:
The graph of the equation is symmetric about both axes. In the first quadrant, the equation is equivalent to . Completing the square gives , so the graph in the first quadrant is an arc of the circle that is centered at and contains the points and . Because is the midpoint of , the arc is a semicircle. The region enclosed by the graph in the first quadrant is the union of isosceles right triangle , where is the origin, and a semicircle with diameter . The triangle and the semicircle have areas and , respectively, so the area of the region enclosed by the graph in all quadrants is .
The problems on this page are the property of the MAA's American Mathematics Competitions