Prove that the linear fractional transformation  maps the circle  into itself. Also prove that  is conformal in .

To prove that the linear fractional transformation \(\Phi(z) = \frac{2z-1}{2-z}\) maps the unit circle \(C: |z| = 1\) into itself, we need to show that if \(|z| = 1\), then \(|\Phi(z)| = 1\).

Let \(z = e^{i\theta}\) be a point on the unit circle, where \(\theta\) is a real parameter.

Then, 
\[
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