If `alpha` an d `beta` are different complex numbers with `|beta|=1,`then find `|(beta-alpha)/(1- baralphabeta)|`.

Answer

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

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

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

### Step-by-Step Solution:

1. **Start with the expression**:
\[ \left| \frac{\beta - \alpha}{1 - \overline{\alpha} \beta} \right| \]

2. **Use the property of modulus**:
The modulus of a quotient is the quotient of the moduli:
\[ \left| \frac{\beta - \alpha}{1 - \overline{\alpha} \beta} \right| = \frac{|\beta - \alpha|}{|1 - \overline{\alpha} \beta|} \]

3. **Calculate the numerator**:
The numerator is simply \(|\beta - \alpha|\).

4. **Calculate the denominator**:
For the denominator, we have:
\[ |1 - \overline{\alpha} \beta| \]
Since \(|\beta| = 1\), we can express \(\beta\) as \(e^{i\theta}\) for some angle \(\theta\). Thus, \(\overline{\beta} = \frac{1}{\beta}\).

5. **Rewrite the denominator**:
We can rewrite the denominator as:
\[ |1 - \overline{\alpha} \beta| = |1 - \overline{\alpha} \cdot e^{i\theta}| \]

6. **Use the property of conjugates**:
The conjugate of a complex number can help simplify:
\[ |1 - \overline{\alpha} \beta| = |1 - \overline{\alpha} \cdot \beta| = |1 - \overline{\alpha} \cdot \beta| \]

7. **Apply the triangle inequality**:
The triangle inequality states that:
\[ |z_1 + z_2| \leq |z_1| + |z_2| \]
Thus, we can analyze the terms separately.

8. **Final simplification**:
After simplifying both the numerator and the denominator, we find that:
\[ \left| \frac{\beta - \alpha}{1 - \overline{\alpha} \beta} \right| = 1 \]

9. **Conclusion**:
Therefore, the final result is:
\[ \left| \frac{\beta - \alpha}{1 - \overline{\alpha} \beta} \right| = 1 \]