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+a0-ib)/(a+a0+ib).
I made a substitution let d = (a+a0+ib) and conj(d) = (a+a0-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.