Let `alpha` and `beta` be two distinct complex numbers, such that `abs(alpha)=abs(beta)`. If real part of `alpha` is positive and imaginary part of `beta` is negative, then the complex number `(alpha+beta)//(alpha-beta)` may be

Options

A. zero
B. real and negative
C. real and positive
D. purely imaginary

Answer

To solve the problem, we need to analyze the given conditions and find the value of the expression \((\alpha + \beta) / (\alpha - \beta)\).

### Step-by-Step Solution:

1. **Understanding the Given Conditions:**
We have two distinct complex numbers \(\alpha\) and \(\beta\) such that \(|\alpha| = |\beta|\). The real part of \(\alpha\) is positive, and the imaginary part of \(\beta\) is negative.

2. **Expressing the Complex Numbers:**
Since \(|\alpha| = |\beta|\), we can express them in polar form:
\[ \alpha = r e^{i\theta} \quad \text{and} \quad \beta = r e^{i\phi} \]
where \(r > 0\) (since \(\alpha\) has a positive real part) and \(\theta\) and \(\phi\) are the angles corresponding to \(\alpha\) and \(\beta\) respectively.

3. **Identifying the Angle Ranges:**
Given that the real part of \(\alpha\) is positive, \(\theta\) must be in the range:
\[ -\frac{\pi}{2} < \theta < \frac{\pi}{2} \]
Since the imaginary part of \(\beta\) is negative, \(\phi\) must be in the range:
\[ -\pi < \phi < 0 \]

4. **Calculating \(\alpha + \beta\) and \(\alpha - \beta\):**
We can now calculate:
\[ \alpha + \beta = r e^{i\theta} + r e^{i\phi} = r(e^{i\theta} + e^{i\phi}) \]
\[ \alpha - \beta = r e^{i\theta} - r e^{i\phi} = r(e^{i\theta} - e^{i\phi}) \]

5. **Finding the Expression \((\alpha + \beta) / (\alpha - \beta)\):**
Now, substituting these into our expression:
\[ \frac{\alpha + \beta}{\alpha - \beta} = \frac{r(e^{i\theta} + e^{i\phi})}{r(e^{i\theta} - e^{i\phi})} = \frac{e^{i\theta} + e^{i\phi}}{e^{i\theta} - e^{i\phi}} \]

6. **Using Euler's Formula:**
Using Euler's formula, we can express \(e^{i\theta}\) and \(e^{i\phi}\) as:
\[ e^{i\theta} = \cos(\theta) + i \sin(\theta) \quad \text{and} \quad e^{i\phi} = \cos(\phi) + i \sin(\phi) \]

7. **Analyzing the Result:**
The numerator \(e^{i\theta} + e^{i\phi}\) and the denominator \(e^{i\theta} - e^{i\phi}\) will yield a complex number. Since \(\theta\) is in the first quadrant (positive real part) and \(\phi\) is in the fourth quadrant (negative imaginary part), the resulting expression will be purely imaginary.

8. **Conclusion:**
Therefore, the expression \((\alpha + \beta) / (\alpha - \beta)\) is purely imaginary.

### Final Answer:
The complex number \((\alpha + \beta) / (\alpha - \beta)\) is purely imaginary.

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