Let z=x+iy be a complex number, where x and y are real numbers. Let A and B be the sets defined by A={z:|z|≤2} and B={z:(1−i)z+(1+i)¯¯¯z≥41} . Find the area of region A∩B.
A
3π−2
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B
3π+2
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C
π+2
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D
π−2
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Solution
The correct option is Cπ−2 z=x+iy A={z:|z|≤2} ⟹√x2+y2≤2 or x2+y2≤4 ⟹z lies on or inside the circle x2+y2=4 B={z:(1−i)z+(1+i)¯¯¯z≥4} ⟹(1−i)(x+iy)+(1+i)(x−iy)≥4 ⟹x+iy−ix+y+x−iy+ix+y≥4 ⟹x+y≥2 Area of region A∩B is the shaded region shown in Fig. 3.19 Area = π(2)24−12×2×2=π−2 Ans: D