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 $x > 0, y > 0$ . $x^2+y^2=x+y$ . $(x - \frac{1}{2})^2+(y - \frac{1}{2})^2=\frac{1}{2}$ . Notice the circle intersect the axe at point $(0, 1)$ and $(1, 0)$ . Find the area of this circle in the first quadrant. The area is made of a semi-circle and a triangle: $\frac{çpi}{4} +\frac{1}{2}$ (Error compiling LaTeX. Unknown error_msg) Because of symmetry, the 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
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All AMC 12 Problems and Solutions

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2016 AMC 12B Problems/Problem 18

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