If `alpha and beta` Are different complex number with `|alpha|=1,` then what is `|(alpha- beta)/(1-alpha beta)|` equal to ?

Options

A. `|beta|`
B. 2
C. 1
D. 0

Explanation

To solve the problem, we need to find the value of

\[ \left| \frac{\alpha - \beta}{1 - \alpha \beta^*} \right| \]

given that \(|\alpha| = 1\) and \(\alpha\) and \(\beta\) are different complex numbers.

### Step-by-step Solution:

1. **Understanding the Modulus Condition**:
Since \(|\alpha| = 1\), we know that \(\alpha \overline{\alpha} = 1\), where \(\overline{\alpha}\) is the conjugate of \(\alpha\).

2. **Expressing the Modulus**:
We can express the modulus of the fraction as:
\[ \left| \frac{\alpha - \beta}{1 - \alpha \beta^*} \right| = \frac{|\alpha - \beta|}{|1 - \alpha \beta^*|} \]

3. **Using the Property of Modulus**:
We know that for any complex number \(z\), \(|z| = |\overline{z}|\). Thus, we can write:
\[ |1 - \alpha \beta^*| = |1 - \overline{\beta} \alpha| \]

4. **Substituting the Conjugate**:
Since \(|\alpha| = 1\), we can also express \(\alpha\) in terms of its conjugate:
\[ |\alpha|^2 = \alpha \overline{\alpha} = 1 \Rightarrow \overline{\alpha} = \frac{1}{\alpha} \]

5. **Finding the Modulus**:
Now, we can rewrite the expression:
\[ \left| \frac{\alpha - \beta}{1 - \alpha \beta^*} \right| = \frac{|\alpha - \beta|}{|1 - \alpha \beta^*|} = \frac{|\alpha - \beta|}{|1 - \overline{\beta} \alpha|} \]

6. **Final Simplification**:
Since \(|\alpha| = 1\), we can conclude that:
\[ \left| \frac{\alpha - \beta}{1 - \alpha \beta^*} \right| = 1 \]

### Conclusion:
Thus, we find that:

\[ \left| \frac{\alpha - \beta}{1 - \alpha \beta^*} \right| = 1 \]