Areas In Circle

areas in circle. A problem

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 Si . Find S1 - S2 + S3 - S4.

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

Solution

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Copyright © 1996-2017 Alexander Bogomolny

areas in circle. A problem

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 Si . Find S1 - S2 + S3 - S4.

Solution

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

areas in circle. A solution

These two lines together with the two axes, split the area of the circle into 9 parts, eight of which cancel out in S1 - S2 + S3 - S4. The only remaining one is the central rectangle whose area is 2a·2b = 4ab.

|Contact| |Front page| |Contents| |Up| |Geometry|

Copyright © 1996-2017 Alexander Bogomolny

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