Question

Mohanlal, a prosperous farmer, has a square land of side 2 km. For the current season, he decides to have some fun. He marks two distinct points on one of the diagonals of the land. Using these points as centers, he constructs two circles. Each of these circles falls completely within the land, and touches at least two sides of the land. To his surprise, the radii of both the circles are exactly equal to $\frac{2}{3}$ km. Mohanlal plants potatoes on the overlapping portion of these circles. Calc

Accepted Answer

D. Square side = 2. Radii $r = 2/3$. Centers are on diagonal, touching sides $\implies$ distance from corner to center $C_1 = r\sqrt{2} = \frac{2\sqrt{2}}{3}$. Distance from opposite corner to $C_2 = \frac{2\sqrt{2}}{3}$. Total diagonal = $2\sqrt{2}$. Distance $C_1 C_2 = 2\sqrt{2} - 2 	imes \frac{2\sqrt{2}}{3} = \frac{2\sqrt{2}}{3}$. Note $C_1 C_2 = r\sqrt{2} = \frac{2\sqrt{2}}{3}$. Intersection of two congruent circles of radius $r$ separated by distance $d = r\sqrt{2}$: The line connecting intersection points $P, Q$ forms a right triangle with $C_1, C_2$ because $C_1 P^2 + C_2 P^2 = r^2 + r^2 = 2r^2 = (r\sqrt{2})^2 = (C_1 C_2)^2$. So angle subtended at center by overlapping arc is $90^\circ$. Area of segment of one circle = Sector area - Triangle area = $\frac{\pi}{4} r^2 - \frac{1}{2} r^2 = r^2 (\frac{\pi - 2}{4})$. Total overlapping area = $2 	imes 	ext{segment area} = 2 	imes \frac

Overlapping Area of Two Circles on Diagonal

Explanation

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