A square is constructed by joining the points A (–1, 1), B (–1, –1), C (1, –1) and D (1, 1). Considering A, B, C and D as centres, four circles each of 2 unit radius are described on the square. Find the area common to these circles.
Text Solution
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Sol. We have to find the area of the shaded region in fig. From the symmetry, Required area = 4 (area of region ORQO) ... (i)
Clearly, R is the point of intersection of the circle having centre at (–1, –1) and radius 2 i.e., (x + 1) 2 + (y +1) 2 = 2 2 with x- axis. Therefore, abscissa of R of given by (x + 1) 2 + (0 + 1) 2 = 2 2 ⇒ x = 
– 1
∴ Area of the region ORQO
= 
=
= 
= 
– 
= – 
+ 1 + 
+ 2 × 
– 
–2 × 
= 
Hence, required area = 4 
sq. units.
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