Let `S=S_1 nn S_2 nn S_3`, where `s_1={z in C :|z|<4}, S_2={z in C :ln[(z-1+sqrt(3)i)/(1-sqrt(31))]>0} and `
`S_3={z in C : Re z > 0}` Area of S=

Options

A. `(10pi)/3`
B. `(20pi)/3`
C. `(16pi)/3`
D. `(32pi)/3`

Answer

B

Explanation

Clearly, `S_(1)` represents the interior of the region bounded by the circle having center at the origin and radius 4 units.

Let `z=x+iy`, then
`(z-1+isqrt(3))/(1-isqrt(3))=(x+iy)/(1-isqrt(3))-1`
`=((x+iy)(1+isqrt(3))/((1-isqrt(3))(1+isqrt(3))-1)) = (x-sqrt(3)y-4)/(4)+i(y+sqrt(3)x)/(4)`
`therefore "Im"(z-1+isqrt(3))/(1-isqrt(3)0 gt 0) rArr ((y+sqrt(3)x)/4) gt 0 rArr y gt -sqrt(3)x`
Thus, `S_(2)` represents the region lying above the line `y=-sqrt(3)x`. Clearly, `S_(3)` represents region representing on the right sides of y-axis.
Hence, `S=S_(1) frown S_(2) frown S_(3)` represents the shaded region shown in Fig.
Area of region S =Area of semi cricel -Area of the sector `OB^(')C`
`=1/2pi(4)^(2)-1/2(4)^(2) xxpi/6 = (20pi)/3` sq. units.