If `|z| =1 ` and `w=(z-1)/(z+1)` (where `z != -1`) then `Re(w)` is

Options

A. 0
B. `1/(|z+1|^2)`
C. `|(1)/(z+1)|.(1)/(|z+1|^2)`
D. `sqrt(2)/(|z+1|^2)`

Answer

A

Explanation

Since `|z| =1 and w = (z-1)/(z+1)`
`rArr z-1 =wz+w rArr =(1+w)/(1-w)rArr |z|=(|1+w|)/(|1-w|)`
`rArr |1-w|=|1+w| " "[ becaue |z|=1]` ltbr. On squaring both sides we get
`1+ |w|^2-2|w| Re (w) =1 + |w|^2+2|w|Re (W)`
[using `|z_1pm z_2|^2=|z_1|^2+|z_2|^2pm 2 |z_1||z_2| Re (bar z_1 z_2)]`
`rArr 4 |w|Re |w|=0`
`rArr " " Re (w)=0`