Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?

Is there anything special with the form:

$$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$
? With $a$ and $z$ are complex numbers.

In fact, I saw it in a problem:

  1. If $|z| = 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| = 1$
  2. If $|z| < 1$ and $|a| < 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| < 1$

I can easily prove the first one with expansion:

$$z=\cos\theta + i\sin \theta \\ a = m +in$$

But it will be terrible to use in the second one. What’s more, I found this form a little special so maybe there is some clever trick without using expansion?

Thank you a lot!

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