If `alpha` and `beta` are different complex number of with `| beta | = 1`, then `|(beta - alpha )/(1- bar(alpha)beta)|` is equal to

Options

A. 0
B. `1//2`
C. 1
D. 2

Answer

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

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

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

### Step-by-Step Solution:

1. **Define T**:
We start by defining \( T \) as follows:
\[ T = \left| \frac{\beta - \alpha}{1 - \overline{\alpha} \beta} \right| \]

2. **Multiply Numerator and Denominator by \(\overline{\beta}\)**:
To simplify \( T \), we multiply both the numerator and denominator by \(\overline{\beta}\):
\[ T = \left| \frac{(\beta - \alpha) \overline{\beta}}{(1 - \overline{\alpha} \beta) \overline{\beta}} \right| \]

3. **Simplify the Numerator**:
The numerator becomes:
\[ (\beta - \alpha) \overline{\beta} = \beta \overline{\beta} - \alpha \overline{\beta} = 1 - \alpha \overline{\beta} \]
(since \(|\beta|^2 = \beta \overline{\beta} = 1\))

4. **Simplify the Denominator**:
The denominator becomes:
\[ (1 - \overline{\alpha} \beta) \overline{\beta} = \overline{\beta} - \overline{\alpha} \beta \overline{\beta} = \overline{\beta} - \overline{\alpha} \]
(again using \(|\beta|^2 = 1\))

5. **Substitute Back into T**:
Now substituting back into \( T \):
\[ T = \left| \frac{1 - \alpha \overline{\beta}}{\overline{\beta} - \overline{\alpha}} \right| \]

6. **Taking the Modulus**:
Since we are interested in the modulus, we can use the property of modulus:
\[ T = \frac{|1 - \alpha \overline{\beta}|}{|\overline{\beta} - \overline{\alpha}|} \]

7. **Recognizing the Reciprocal**:
Notice that the expression for \( T \) can be rewritten as:
\[ T = \frac{|1 - \alpha \overline{\beta}|}{|\overline{\beta} - \overline{\alpha}|} = \frac{1}{T} \]
This implies:
\[ T^2 = 1 \]

8. **Final Value of T**:
Since \( T \) is a modulus, it must be non-negative. Therefore, we conclude:
\[ T = 1 \]

### Conclusion:
Thus, the value of \( \left| \frac{\beta - \alpha}{1 - \overline{\alpha} \beta} \right| \) is equal to \( 1 \).