Let C be the circle center (0, 0), radius 1. Let X, Y be two points on C with positive x and y coordinates. Let X1, Y1 be the points on the x-axis with the same x-coordinates as X and Y respectively, and let X2, Y2 be the points on the y-axis with the same y-coordinates. Show that the area of the region XYY1X1 plus the area of the region XYY2X2 depends only on the length of the arc XY, and not on its position.
Solution
Easy.
Let O the origin. Area XYY1X1 = area sector OXY + area OYY1 - area OXX1, and area XYY2X2 = area sector OXY + area OXX2 - area OYY2. But area OXX1 = area OXX2 and area OYY1 = area OYY2. Hence area XYY1X1 + area XYY2X2 = 2 area sector OXY, which is independent of the position of the arc.
© John Scholes
jscholes@kalva.demon.co.uk
12 Dec 1998