If `a gt 0 and z=(1+i)^2/(a- i)` has magnitude `sqrt(2/5) then barz is ` equal to

Options

A. `1/5-3/5 i`
B. `-1/5-3/5 i `
C. `1/5+3/5 i `
D. `-3/5-1/5 i `

Explanation

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

### Step-by-Step Solution:

1. **Calculate \( (1+i)^2 \)**:
\[ (1+i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i - 1 = 2i \]
Hence, \( z = \frac{2i}{a-i} \).

2. **Rationalize the denominator**:
To rationalize \( z \), multiply the numerator and denominator by the conjugate of the denominator:
\[ z = \frac{2i}{a-i} \cdot \frac{a+i}{a+i} = \frac{2i(a+i)}{(a-i)(a+i)} = \frac{2i(a+i)}{a^2 + 1} \]

3. **Simplify the expression**:
Expanding the numerator:
\[ z = \frac{2ai + 2i^2}{a^2 + 1} = \frac{2ai - 2}{a^2 + 1} = \frac{-2 + 2ai}{a^2 + 1} \]
Thus, we can express \( z \) as:
\[ z = \frac{-2}{a^2 + 1} + \frac{2a}{a^2 + 1}i \]

4. **Identify the real and imaginary parts**:
Let \( x = \frac{-2}{a^2 + 1} \) (real part) and \( y = \frac{2a}{a^2 + 1} \) (imaginary part).

5. **Calculate the magnitude of \( z \)**:
The magnitude \( |z| \) is given by:
\[ |z| = \sqrt{x^2 + y^2} = \sqrt{\left(\frac{-2}{a^2 + 1}\right)^2 + \left(\frac{2a}{a^2 + 1}\right)^2} \]
\[ = \sqrt{\frac{4}{(a^2 + 1)^2} + \frac{4a^2}{(a^2 + 1)^2}} = \sqrt{\frac{4(1 + a^2)}{(a^2 + 1)^2}} = \frac{2\sqrt{1 + a^2}}{a^2 + 1} \]

6. **Set the magnitude equal to \( \sqrt{\frac{2}{5}} \)**:
\[ \frac{2\sqrt{1 + a^2}}{a^2 + 1} = \sqrt{\frac{2}{5}} \]

7. **Square both sides**:
\[ \frac{4(1 + a^2)}{(a^2 + 1)^2} = \frac{2}{5} \]

8. **Cross-multiply**:
\[ 20(1 + a^2) = 2(a^2 + 1)^2 \]

9. **Expand and simplify**:
\[ 20 + 20a^2 = 2(a^4 + 2a^2 + 1) \]
\[ 20 + 20a^2 = 2a^4 + 4a^2 + 2 \]
\[ 2a^4 - 16a^2 - 18 = 0 \]

10. **Divide by 2**:
\[ a^4 - 8a^2 - 9 = 0 \]

11. **Let \( u = a^2 \)**:
\[ u^2 - 8u - 9 = 0 \]

12. **Use the quadratic formula**:
\[ u = \frac{8 \pm \sqrt{64 + 36}}{2} = \frac{8 \pm 10}{2} \]
\[ u = 9 \quad \text{or} \quad u = -1 \quad (\text{discard since } a^2 > 0) \]
Thus, \( a^2 = 9 \) implies \( a = 3 \).

13. **Find \( z \)**:
Substitute \( a = 3 \) back into \( z \):
\[ z = \frac{-2 + 6i}{10} = -\frac{1}{5} + \frac{3}{5}i \]

14. **Find the conjugate \( \bar{z} \)**:
The conjugate of \( z \) is:
\[ \bar{z} = -\frac{1}{5} - \frac{3}{5}i \]

### Final Answer:
\[ \bar{z} = -\frac{1}{5} - \frac{3}{5}i \]