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## Homework Statement

Let a be a complex number for which Im(a) ≠ 0, and f(z) = (z + conj(a))/(z + a).

Prove f(z) maps the real axis onto the circle lwl = 1.

**2. The attempt at a solution**

I wrote out f(z) in an a+bi for and then with the Im(a) ≠ 0 I set the equation as

f(a+bi) = (a+a

_{0}-ib)/(a+a

_{0}+ib).

I made a substitution let d = (a+a

_{0}+ib) and conj(d) = (a+a

_{0}-ib)

This gave me d/conj(d). I have exhausted all of the identities I could remember/find and I see no path leading this line of thinking to a circle.