If `alpha, beta` are real, what is `|(alpha+ibeta)/(beta+ialpha)|` equal to ?

Options

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

Explanation

To solve the problem \( \left| \frac{\alpha + i\beta}{\beta + i\alpha} \right| \), we will follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ z = \frac{\alpha + i\beta}{\beta + i\alpha} \]

### Step 2: Use the property of modulus
The property of modulus states that:
\[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \]
Applying this property, we have:
\[ \left| z \right| = \frac{|\alpha + i\beta|}{|\beta + i\alpha|} \]

### Step 3: Calculate the modulus of the numerator
The modulus of the numerator \( |\alpha + i\beta| \) is calculated as:
\[ |\alpha + i\beta| = \sqrt{\alpha^2 + \beta^2} \]

### Step 4: Calculate the modulus of the denominator
The modulus of the denominator \( |\beta + i\alpha| \) is calculated as:
\[ |\beta + i\alpha| = \sqrt{\beta^2 + \alpha^2} \]

### Step 5: Substitute the moduli into the expression
Now substituting the moduli into our expression for \( |z| \):
\[ |z| = \frac{\sqrt{\alpha^2 + \beta^2}}{\sqrt{\beta^2 + \alpha^2}} \]

### Step 6: Simplify the expression
Since \( \alpha^2 + \beta^2 = \beta^2 + \alpha^2 \), we can simplify:
\[ |z| = \frac{\sqrt{\alpha^2 + \beta^2}}{\sqrt{\alpha^2 + \beta^2}} = 1 \]

### Conclusion
Thus, we find that:
\[ \left| \frac{\alpha + i\beta}{\beta + i\alpha} \right| = 1 \]