Let the curve z(1+i)+ z (1-i)=4, z C , divide the region |z-3| 1 into two parts of areas and . Then | - | equals :

Step-by-step solution

aligned & Let z=x+i y & (x+i y)(1+i)+(x-i y)(1-i)=4 & x+i x+i y-y+x-i x-i y-y=4 & 2 x-2 y=4 & x-y=2 & |z-3| 1 & (x-3)^2+y^2 1 aligned Area of shaded region = 1^2 4 - 1 2 1 1= 4 - 1 2 Area of unshaded region inside the circle = 3 4 1^2+ 1 2 1 1= 3 4 + 1 2 difference of area = ( 3 4 + 1 2 )- ( 4 - 1 2 )= 2 +1