Let S=S₁ S₂ S₃ where: aligned & S₁= z|z C,| z 0 & S₃= z z C, R(z)0 aligned If the area of S can be expressed as a b , where a and b are positive integers that are relatively prime. Find a+b .

Question

Let S=S₁ S₂ S₃ where: aligned & S₁= z|z C,| z 0 & S₃= z z C, R(z)>0 aligned If the area of S can be expressed as a b , where a and b are positive integers that are relatively prime. Find a+b .

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Correct answer: 23

Let z=x+i yS₁ denotes the interior of circle of radius 4 units S₃ denotes x>0S₂= Im ( (x-1+i(y+ 3 ))(1+i 3 ) 4 )>0S₂=i(y+ 3 x)>0 Now the shaded region represents the required area Required area = Area of quarter of circle + Area of sector = r^2 4 + r^2 360^ =4 + 8 3 20 3

Let S=S₁ S₂ S₃ where: aligned & S₁= z|z C,| z 0 & S₃= z z C, R(z)>0 aligned If the area of S can be expressed as a b , where a and b are positive integers that are relatively prime. Find a+b .

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