Let Tz ptz q lambda z pz q be the normal form of a Mobi

Let Tz - p/tz - q = lambda z - p/z - q be the normal form of a Mobius transformation with two fixed points. Prove that lambda = (Tz, z, p, q), where z is any complex number.

Solution

Given that (Tz-p)/(Tz-q)=lambda (z-p)/(z-q)

-----> lambda =(Tz-p)(z-q)/(z-p)(Tz-q)

And which is the cross ratio of Tz,z,p,q

Hence lambda=(Tz,z,p,q).