Mobius Maps: Parallel and Perpendicular Lines, Disjoint Circles

[Sammicalvin]

Homework Statement

Suppose T is a Mobius map take Real -> Real and infinite infinitely to 0
a) What's the image of the family of lines parallel to Real?
b) What's the image of the family of lines perpendicular to Real?
c) Show the Mobius map take D = {z :|z|<1} onto itself iff Tz = e^i(theta) * (z-a)/(1-az)
for a belongs to D and theta belongs to Real
d) C1 and C2 are disjoint circles in Complex . Show there's a Mobius map s.t. TC1 and TC2 are concentric

Homework Equations

The Attempt at a Solution

a) Since the mobius map take infinitely to 0, we are expect to get a circle through 0?
b) Since the mobius map take infinitely to 0, we are expect to get a circle through 0 and perpendicular to Real? (since mobius map preserves angle)?
c) let z = e^iT ?
d) i hv no idea

 

[Socrates' Pen]

Neat questions. I don't have a clue either.
Anyone want to help?