All the points in the set `S={(alpha+i)/(alpha-i):alpha in R } (i= sqrt(-1))lie` on a

Options

A. straight line whose slope is 1
B. circle whose radius is 1
C. straight line whose slope is `–1`
D. circle whose radius is `sqrt(2)`

Explanation

To solve the problem, we need to analyze the set \( S = \left\{ \frac{\alpha + i}{\alpha - i} : \alpha \in \mathbb{R} \right\} \) and determine the geometric representation of the points in this set.

### Step-by-Step Solution:

1. **Define the Expression**:
Let \( z = \frac{\alpha + i}{\alpha - i} \).

2. **Multiply by the Conjugate**:
To simplify \( z \), multiply the numerator and denominator by the conjugate of the denominator:
\[ z = \frac{(\alpha + i)(\alpha + i)}{(\alpha - i)(\alpha + i)} = \frac{(\alpha + i)^2}{\alpha^2 + 1} \]

3. **Expand the Numerator**:
Expand the numerator:
\[ (\alpha + i)^2 = \alpha^2 + 2\alpha i - 1 \quad (\text{since } i^2 = -1) \]
Therefore, we have:
\[ z = \frac{\alpha^2 - 1 + 2\alpha i}{\alpha^2 + 1} \]

4. **Separate Real and Imaginary Parts**:
From the expression for \( z \), we can identify the real part \( x \) and the imaginary part \( y \):
\[ x = \frac{\alpha^2 - 1}{\alpha^2 + 1}, \quad y = \frac{2\alpha}{\alpha^2 + 1} \]

5. **Calculate \( x^2 + y^2 \)**:
To find the modulus \( |z| \), we calculate \( x^2 + y^2 \):
\[ x^2 + y^2 = \left(\frac{\alpha^2 - 1}{\alpha^2 + 1}\right)^2 + \left(\frac{2\alpha}{\alpha^2 + 1}\right)^2 \]

6. **Combine the Fractions**:
Combine the fractions:
\[ x^2 + y^2 = \frac{(\alpha^2 - 1)^2 + (2\alpha)^2}{(\alpha^2 + 1)^2} \]
Expanding the numerator:
\[ (\alpha^2 - 1)^2 + (2\alpha)^2 = \alpha^4 - 2\alpha^2 + 1 + 4\alpha^2 = \alpha^4 + 2\alpha^2 + 1 = (\alpha^2 + 1)^2 \]

7. **Simplify**:
Thus, we have:
\[ x^2 + y^2 = \frac{(\alpha^2 + 1)^2}{(\alpha^2 + 1)^2} = 1 \]

8. **Conclusion**:
Since \( x^2 + y^2 = 1 \), this implies that the modulus \( |z| = 1 \). Therefore, all points \( z \) lie on the unit circle in the complex plane.

### Final Result:
The points in the set \( S \) lie on a circle with center at \( (0, 0) \) and radius \( 1 \).