The wording is not entirely clear in this exercise. Somebody who posted a solution in the discussion forum was unsure what is actually asked. My guess is that to find a Mobius transformation that rotates the z-plane unit circle into the t-plane real axis.

A Mobius transformation

can be re-written as

. So, one needs three linear equations for the unknown

,

and

.

Again, there're ambiguities. Any transformation that rotates the north pole to any point on the unit circle will do. Moreover, one can freely choose the direction of the t-plane real axis. Two degrees of freedom, so it seems.

I tested and knew that the following two choices will provide the answer Penrose asked for:

(rotation of north pole to -1)

(direction of the t-plane real axis)

It also follows from the first choice that

Substitute these three

and

to the Mobius transformatioin to get three linear equations whose solutions are

and

.

Finally,

. The reverse correspondence,

dependent on

is straightforward.