If Re`((z-1)/(z+1))=0` where `z=x+iy` is a complex number, then which one of the following is correct? (a) `z=1+i` (b) `|z|= 2`. (c) `z=1-i`

Options

A. `z=1=i`
B. `|z|=2`
C. `z=1-i`
D. `|z|=1`

Answer

D

Explanation

`(z-1)/(z+1)=(x+iy-1)/(x=iy+1)`
`(z-1)/(z+1)=(x^(2)+y^(2)-1+2iy)/(x^(2)+y^(2)+2x+1)`
`impliesRe((z-1)/(z+1))=(x^(2)+y^(2)-1)/(x^(2)+y^(2)+2x+1)=0`
`impliesx^(2)+y^(2)-1=0`
`impliesx^(2)+y^(2)=1`
Also, `z barz=x^(2)+y^(2)=1`
`and z barz=|z|^(2)`
`implies|z|^(1)=1`
`implies|z|=1`