Let `z=x+i ydot` Then find the locus of `P(z)` such that `(1+ z )/z in Rdot`

Options

A. union of lines with equations `x=0` and `y=-1//2`but excluding origin.
B. union of lines with equations `x=0` and `y=1//2`but excluding origin.
C. union of lines with equations `x=-1//2` and `y=0`but excluding origin.
D. union of lines with equations `x=1//2` and `y=0`but excluding origin.

Answer

C

Explanation

`(c )` Given `(1+barz)/(z)` is real `implies(1+barz)/(z)=(1+z)/(z)`
`impliesbarz+barz^(2)=z+z^(2)implies(barz-z)+(barz-z)(barz+z)=0`
`implies(barz-z)(1+barz+z)=0`
So either `barz=z(z ne 0)` or `z+barz+1=0`
`implies y=0` or `x=(-1)/(2)` but excluding origin.