Intersections of a Circle with the Four QuadrantsThe Ariel University Center in Samaria (Israel) runs an online mathematical olympiad for college students. During the 2009-2010 season they offered several beautiful problems, one of which might appear harder for college students than for younger mathematicians:
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 j-th quadrant (j = 1, 2, 3, 4) by Sj . Find The applet below illustrates the problem. (The circle is draggable. Its radius could be changed with the scrollbar at the bottom of the applet.)
 |Activities| |Contact| |Front page| |Contents| |Geometry| |Proofs| |Store| Copyright © 1996-2012 Alexander Bogomolny
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 j-th quadrant (j = 1, 2, 3, 4) by Sj . Find
SolutionDraw two additional lines x = 2a and y = 2b. Together with the axes, the lines split the circle into four areas: four curvilinear rectangles, or cut-off circular segments (equal two-by-two), four curvilinear triangles (all of which are equal - by symmetry), and a rectangle in the middle centered at point P. In the (algebraic) sum |Activities| |Contact| |Front page| |Contents| |Geometry| |Proofs| |Store| Copyright © 1996-2012 Alexander Bogomolny |
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