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Question

# ABCD is a square with side a. With centres A, B, C and D four circles are drawn such that each circle touches externally two of the remaining three circles. Let δ be the area of the region in the interior of the square and exterior of the circles. Then the maximum value of δ is :

A
a2(1π)
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B
a2(4π4)
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C
a2(π1)
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D
πa24
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Solution

## The correct option is B a2(4−π4)Let ABCD is a square with side a.∴A(□ABCD)=a2 Let's draw four circles with centre A,B,C and D such that each circle touches two of the remaining circles externally.As the circles drawn, their radius will be r=a/2 and one fourth part of a circle lie in the square.Area of a each circle =πr2=a24πSo, for each circle the area covered by the rectangle =a2π44=a2π16.∴ the area covered by four circles =4×a2π16=a2π4.∴ δ=A(□ABCD)−(Area of □ABCD covered by 4 circle) =a2−a2π4=4a2−a2π4 =a2(4−π4)∴ δ=a2(4−π4)

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