If the point `z_(1) = 1+ i`, where `i = sqrt( -1)` is the reflection of a point `z_(2) = x+iy` in the line `ibar(z)-iz = 5`, then the point `z_(2)` is

Options

A. A. `1+4i`
B. B. `4+1`
C. C. `1-i`
D. D. `-1-i`

Answer

To solve the problem, we need to find the point \( z_2 = x + iy \) given that \( z_1 = 1 + i \) is the reflection of \( z_2 \) in the line defined by the equation \( i \bar{z} - iz = 5 \).

### Step-by-Step Solution:

1. **Identify the line equation**:
The given line is \( i \bar{z} - iz = 5 \). We can express \( z \) as \( z = x + iy \) and \( \bar{z} = x - iy \).
\[ i(x - iy) - i(x + iy) = 5 \]

2. **Substitute \( z \) and \( \bar{z} \)**:
Substitute \( z \) and \( \bar{z} \) into the line equation:
\[ i(x - iy) - i(x + iy) = 5 \]
Simplifying this gives:
\[ ix + y - ix - y = 5 \]
This simplifies to:
\[ 2y = 5 \quad \Rightarrow \quad y = \frac{5}{2} \]

3. **Determine the y-coordinate of \( z_2 \)**:
We know that the reflection \( z_1 = 1 + i \) has a y-coordinate of 1. The line \( y = \frac{5}{2} \) is above this point. The distance from \( z_1 \) to the line is:
\[ \text{Distance} = \frac{5}{2} - 1 = \frac{3}{2} \]

4. **Find the y-coordinate of \( z_2 \)**:
Since \( z_2 \) is below the line, we subtract this distance from the line's y-coordinate:
\[ y_2 = \frac{5}{2} - \frac{3}{2} = 1 \]

5. **Determine the x-coordinate of \( z_2 \)**:
The x-coordinate of \( z_2 \) remains the same as that of \( z_1 \) because reflections across a horizontal line maintain the x-coordinate. Therefore:
\[ x_2 = 1 \]

6. **Combine to find \( z_2 \)**:
Thus, the coordinates of \( z_2 \) are:
\[ z_2 = 1 + \frac{5}{2}i \]

### Final Answer:
The point \( z_2 \) is:
\[ z_2 = 1 + 4i \]