If z = `((1+i)^(2))/(alpha-i)`,` alpha in R ` has magnitude `sqrt((2)/(5))` then the value of `alpha` is (i) 3 only (ii) `-3` only (iii) 3 or -3 (iv) none of these

Options

A. 3 only
B. `-3` only
C. 3 or -3
D. none of these

Answer

To solve the problem, we need to find the value of \( \alpha \) given that the magnitude of \( z \) is \( \sqrt{\frac{2}{5}} \) and \( z = \frac{(1+i)^2}{\alpha - i} \).

### Step-by-Step Solution:

1. **Calculate \( (1+i)^2 \)**:
\[ (1+i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i \]

2. **Set up the equation for the magnitude**:
We know that:
\[ |z| = \left| \frac{(1+i)^2}{\alpha - i} \right| = \frac{|(1+i)^2|}{|\alpha - i|} \]
Given that \( |z| = \sqrt{\frac{2}{5}} \), we can write:
\[ \frac{|(1+i)^2|}{|\alpha - i|} = \sqrt{\frac{2}{5}} \]

3. **Calculate \( |(1+i)^2| \)**:
\[ |(1+i)^2| = |2i| = 2 \]

4. **Substitute the magnitude into the equation**:
\[ \frac{2}{|\alpha - i|} = \sqrt{\frac{2}{5}} \]

5. **Cross-multiply**:
\[ 2 = |\alpha - i| \cdot \sqrt{\frac{2}{5}} \]

6. **Solve for \( |\alpha - i| \)**:
\[ |\alpha - i| = \frac{2}{\sqrt{\frac{2}{5}}} = 2 \cdot \sqrt{\frac{5}{2}} = \sqrt{20} = 2\sqrt{5} \]

7. **Express \( |\alpha - i| \) in terms of \( \alpha \)**:
\[ |\alpha - i| = \sqrt{\alpha^2 + 1} \]
So we have:
\[ \sqrt{\alpha^2 + 1} = 2\sqrt{5} \]

8. **Square both sides**:
\[ \alpha^2 + 1 = (2\sqrt{5})^2 \]
\[ \alpha^2 + 1 = 4 \cdot 5 = 20 \]

9. **Solve for \( \alpha^2 \)**:
\[ \alpha^2 = 20 - 1 = 19 \]

10. **Find \( \alpha \)**:
\[ \alpha = \pm \sqrt{19} \]

### Conclusion:
Since \( \sqrt{19} \) does not match any of the provided options, we conclude that the answer is:
- **(iv) none of these**