Let `z=((1+i)^(2))/(a-i)(agta) and |z|=sqrt((2)/(5))`, then `overline(z)` is equal to

Options

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

Answer

To solve the problem step by step, we need to simplify the expression for \( z \) and then find its conjugate. Let's break it down:

### Step 1: Simplify \( z \)

Given:
\[ z = \frac{(1+i)^2}{a-i} \]

First, we simplify \( (1+i)^2 \):
\[ (1+i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i - 1 = 2i \]

Thus, we have:
\[ z = \frac{2i}{a-i} \]

### Step 2: Find the modulus of \( z \)

We know that:
\[ |z| = \frac{|2i|}{|a-i|} \]

Calculating the moduli:
- \( |2i| = 2 \)
- \( |a-i| = \sqrt{a^2 + (-1)^2} = \sqrt{a^2 + 1} \)

So, we can write:
\[ |z| = \frac{2}{\sqrt{a^2 + 1}} \]

### Step 3: Set the modulus equal to the given value

We are given that:
\[ |z| = \sqrt{\frac{2}{5}} \]

Setting the two expressions for \( |z| \) equal gives:
\[ \frac{2}{\sqrt{a^2 + 1}} = \sqrt{\frac{2}{5}} \]

### Step 4: Cross-multiply and simplify

Cross-multiplying yields:
\[ 2 \sqrt{5} = \sqrt{2} \cdot \sqrt{a^2 + 1} \]

Squaring both sides results in:
\[ 4 \cdot 5 = 2(a^2 + 1) \]
\[ 20 = 2a^2 + 2 \]
\[ 18 = 2a^2 \]
\[ a^2 = 9 \]
\[ a = 3 \quad (\text{since } a > 0) \]

### Step 5: Substitute \( a \) back into \( z \)

Now substituting \( a = 3 \) back into the expression for \( z \):
\[ z = \frac{2i}{3-i} \]

### Step 6: Rationalize the denominator

To rationalize \( z \), multiply the numerator and denominator by the conjugate of the denominator:
\[ z = \frac{2i(3+i)}{(3-i)(3+i)} = \frac{6i + 2i^2}{9 + 1} = \frac{6i - 2}{10} = \frac{-2}{10} + \frac{6i}{10} = -\frac{1}{5} + \frac{3i}{5} \]

### Step 7: Find the conjugate of \( z \)

The conjugate of \( z \), denoted \( \overline{z} \), is obtained by changing the sign of the imaginary part:
\[ \overline{z} = -\frac{1}{5} - \frac{3i}{5} \]

### Final Answer

Thus, the conjugate \( \overline{z} \) is:
\[ \overline{z} = -\frac{1}{5} - \frac{3i}{5} \]

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