If `z_(1)`, `z_(2)` are complex numbers such that `Re(z_(1))=|z_(1)-2|`, `Re(z_(2))=|z_(2)-2|` and `arg(z_(1)-z_(2))=pi//3` , then `Im(z_(1)+z_(2))=`

Options

A. `2//sqrt(3)`
B. `4//sqrt(3)`
C. `2//sqrt(3)`
D. `sqrt(3)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the complex numbers
Let \( z_1 = x_1 + i y_1 \) and \( z_2 = x_2 + i y_2 \), where \( x_1, y_1, x_2, y_2 \) are real numbers representing the real and imaginary parts of \( z_1 \) and \( z_2 \) respectively.

### Step 2: Set up the equations from the given conditions
From the problem, we have:
1. \( \text{Re}(z_1) = |z_1 - 2| \)
2. \( \text{Re}(z_2) = |z_2 - 2| \)

Substituting \( z_1 \) and \( z_2 \) into these equations, we get:
- \( x_1 = |(x_1 - 2) + i y_1| \)
- \( x_2 = |(x_2 - 2) + i y_2| \)

### Step 3: Calculate the modulus
The modulus can be expressed as:
- \( |z_1 - 2| = \sqrt{(x_1 - 2)^2 + y_1^2} \)
- \( |z_2 - 2| = \sqrt{(x_2 - 2)^2 + y_2^2} \)

Thus, we have:
1. \( x_1 = \sqrt{(x_1 - 2)^2 + y_1^2} \)
2. \( x_2 = \sqrt{(x_2 - 2)^2 + y_2^2} \)

### Step 4: Square both sides to eliminate the square root
Squaring both sides of the equations gives:
1. \( x_1^2 = (x_1 - 2)^2 + y_1^2 \)
2. \( x_2^2 = (x_2 - 2)^2 + y_2^2 \)

Expanding these:
1. \( x_1^2 = x_1^2 - 4x_1 + 4 + y_1^2 \)
- This simplifies to: \( 4x_1 - 4 + y_1^2 = 0 \) or \( y_1^2 = 4x_1 - 4 \)
2. \( x_2^2 = x_2^2 - 4x_2 + 4 + y_2^2 \)
- This simplifies to: \( 4x_2 - 4 + y_2^2 = 0 \) or \( y_2^2 = 4x_2 - 4 \)

### Step 5: Relate \( y_1 \) and \( y_2 \)
From the equations derived, we have:
- \( y_1 = \sqrt{4x_1 - 4} \)
- \( y_2 = \sqrt{4x_2 - 4} \)

### Step 6: Use the argument condition
We know that \( \arg(z_1 - z_2) = \frac{\pi}{3} \). This implies:
\[ \frac{y_1 - y_2}{x_1 - x_2} = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Step 7: Rearranging the equation
This gives us:
\[ y_1 - y_2 = \sqrt{3}(x_1 - x_2) \]

### Step 8: Substitute \( y_1 \) and \( y_2 \)
Substituting \( y_1 \) and \( y_2 \) into the equation:
\[ \sqrt{4x_1 - 4} - \sqrt{4x_2 - 4} = \sqrt{3}(x_1 - x_2) \]

### Step 9: Solve for \( y_1 + y_2 \)
We can express \( y_1 + y_2 \) using the previous equations:
\[ y_1 + y_2 = \sqrt{4x_1 - 4} + \sqrt{4x_2 - 4} \]

### Step 10: Find \( y_1 + y_2 \)
From the earlier derived equations, we can find that:
\[ y_1 + y_2 = \frac{4}{\sqrt{3}} \]

### Final Answer
Thus, the imaginary part of \( z_1 + z_2 \) is:
\[ \text{Im}(z_1 + z_2) = y_1 + y_2 = \frac{4}{\sqrt{3}} \]