Let `S=S_1 cap S_2 cap S_3` where
`S_1={z in C:|z| lt 4"}",S_2={z in C: lm[(z-1+sqrt(3)i)/(1-sqrt(3)i)]gt0}` and `S_3:{z in : Re z lt 0 }`
Let z be any point in `A cap B cap C`
The `|z+1-i|^2+|z-5-i|^2` lies between

Options

A. 25 and 29
B. 30 and 34
C. 35 and 39
D. 40 and 44

Answer

C

Explanation

Clearly, P(z) is the only poin in `A frown B frown C` satisfying `|z-(2+i)|=3`. We observe that `z_(1)=-1+i` and `z_(2)=5+i` both lie on the circle `|z-(2+i)|=3`, and their mid-point is the center of the circle. So, `A(z_(1))`and `B(z_(2))` are the end-points of a diameter of the circle `|z-(2+i)|=3`
`|z+1-i|^(2)+|z-5-i|^(2)=QA^(2)+QB^(2)=AB^(2)=36`