Transmission Line Length for Real Input Impedance?

[sandy.bridge]

Homework Statement

Hey guys. I have a transmission line with a known characteristic impedance (50 Ω) of length L that is connected to an antenna that has a complex impedance of (50-j25)Ω.

I am having extreme difficulty determining the minimum length (expressed in wavelengths) of line such that the input impedance is real.

Let τ denote the reflection coefficient.

I know that for the input impedance to have no complex component that (1+τ*e^(-jx))/(1-τ*e^(-jx)) should result in complex components vanishing. However, this is a rather complicated formula to solve in the state that it is in because both e^(-jx) and τ are complex numbers.

Am I missing something here?

 

[sandy.bridge]

Fortunately I was able to answer my own question. Thanks anyway!