# Areas In Circle

The point P = P(a, b) is located in the first quadrant. Consider a circle centered at the point P with radius greater than √a² + b², and denote the area of the part of this circle located in the i-th quadrant (i = 1, 2, 3, 4) by S_{i} . Find S_{1} - S_{2} + S_{3} - S_{4}.

This problem has been included in the 2009-2010 international online mathematical olympiad run by the Ariel University Center in Samaria (Israel).

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Copyright © 1996-2018 Alexander Bogomolny

The point P = P(a, b) is located in the first quadrant. Consider a circle centered at the point P with radius greater than √a² + b², and denote the area of the part of this circle located in the i-th quadrant (i = 1, 2, 3, 4) by S_{i} . Find S_{1} - S_{2} + S_{3} - S_{4}.

### Solution

Draw two extra lines, x = 2a and y = 2b:

These two lines together with the two axes, split the area of the circle into 9 parts, eight of which cancel out in S_{1} - S_{2} + S_{3} - S_{4}. The only remaining one is the central rectangle whose area is

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Copyright © 1996-2018 Alexander Bogomolny