Question

`z` and `w` are two non-zero complex numbers such that `abs(z)=abs(w)` and `Arg(z) + Arg(w) = pi` then find `z` ?

Answer 1

`z = -bar(w)" "` for any `w` with `Arg(w) in [0, pi]`

Explanation:

If `w = r(cos theta + i sin theta)` with `theta in [0, pi]`

then:

`z = r(cos (pi-theta) + i sin (pi-theta)) = r(-cos(theta)+i sin(theta)) = -bar(w)`

gives: `Arg(z) + Arg(w) = (pi-theta) + theta = pi`

If `w = r(cos theta + i sin theta)` with `theta in (-pi, 0)`

then since `Arg(z) in (-pi, pi]` there is no value of `z` which will give `Arg(z) + Arg(w) = pi`.