linear fractional maps and geodesics

[jenniferwu3232]

Consider the fractional linear map Φ(z)=(az+b/cz+d) where a, b, c d are real numbers with ad-bc=1. Suppose in addition that |a+d| > 2

1. show that if c ≠ 0, there exists exactly 2 points x ∈R such that Φ (x)=x. Hint: quadratic formula
2. use part a) to show that there is a unique complete geodesic g in H^2 such that that Φ (g)=g

I have figured out part a) but i am stuck on part b). I know that a complete geodesic curves is in H^2 are the upper half vertical lines starting on the x-axis and the upper half circles centered on the x-axis, but I don't know how to apply this fact.