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}`

Options

A. `(2-sqrt(3))/(2)`
B. `(2+sqrt(3))/(2)`
C. `(3-sqrt(3))/(2)`
D. `(3+sqrt(3))/(2)`

Answer

C

Explanation

Clearly, `|1-3i-z|=|z-(1-3i)|`= Distance between z and `(1,-3)`. Thus, `|1-3i-z|` represents distance of any point in the region minimum by S from the point `P(1,-3)`. So `|1-3i-z|` is minimum when z is the foot of the perpendicular from k`kP(1,-3)` on the line `y=-sqrt(3)x`. (see fig)
Hence, the minimum value of `|1-3i-z|` is equal to the length of perpendicular drawn from `P(1,-3)` on the line `sqrt(3x)+y=0` and is given by
`|(sqrt(3)-3)/(sqrt(3)^(2)+1^(2))|=(3-sqrt(3))/(2)`