2016 AMC 12B Problems/Problem 18

Problem

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$

$\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$

Solution

Consider the case when $x > 0, y > 0.$ $x^2+y^2=x+y.$ $(x - 0.5)^2+(y - 0.5)^2=0.5.$ Find the area of this circle in the first quadrant. Notice the circle intersect the axe at point $(0, 1) and (1, 0).$ : $0.5 + 0.25\pi$ Because of symmetry, that area is the same in all four quadrants. The answer is $\boxed{\textbf{(B)}\ 2 + \pi}$

2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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2016 AMC 12B Problems/Problem 18

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