If `z_1` is a complex number other than -1 such that `|z_1|=1\ a n d\ z_2=(z_1-1)/(z_1+1)` , then the real parts of `z_2` is .

Options

A. 0
B. 1
C. Real part of `z_1`
D. None of these

Answer

A

Explanation

`abs(Z_1) = 1` and `Z_2 = frac{Z_1 − 1} {z_1 + 1}`

Let us assume `z_1=x+iy`

`abs(Z_1)=1`

`sqrt(x^2+y^2)=1`

`x^2+y^2=1 ..........(1)`

`Z_2=frac{z_1-1}{z_1+1}`

`Z_2=frac{(x-1)+iy}{(x+1)+iy}xx frac{(x-1)+iy}{(x-1)+iy}`

`Z_2=frac{x^2-1+i(-xy+y+xy+y)-i^2y^2}{x^2+2x+1-i^2y^2}`
...