2002 AMC 12B Problems/Problem 25
Contents
Problem
Let , and let denote the set of points in the coordinate plane such that The area of is closest to
Solution 1
The first condition gives us that
which is a circle centered at with radius . The second condition gives us that
Thus either
or
Each of those lines passes through and has slope , as shown above. Therefore, the area of is half of the area of the circle, which is .
Solution 2
Similar to Solution 1, we proceed to get the area of the circle satisfying , or .
Since , we have that by symmetry, if is in , then is not, and vice versa. Therefore, the shaded part of the circle above the line has the same area as the unshaded part below , and the unshaded part above has the same area as the shaded part below . This means that exactly half the circle is shaded, allowing us to divide by two to get . ~samrocksnature + ddot1
See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
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