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 step by step, we will start by defining the complex numbers \( z_1 \) and \( z_2 \) in terms of their real and imaginary parts.

### Step 1: Define the complex numbers
Let:
\[ z_1 = x_1 + i y_1 \]
\[ z_2 = x_2 + i y_2 \]

### Step 2: Set up the equations based on the given conditions
From the problem, we have:
1. \( \text{Re}(z_1) = |z_1 - 2| \)
2. \( \text{Re}(z_2) = |z_2 - 2| \)
3. \( \arg(z_1 - z_2) = \frac{\pi}{3} \)

### Step 3: Analyze the first condition
For \( z_1 \):
\[ \text{Re}(z_1) = x_1 \]
\[ |z_1 - 2| = |(x_1 - 2) + i y_1| = \sqrt{(x_1 - 2)^2 + y_1^2} \]
Setting these equal gives:
\[ x_1 = \sqrt{(x_1 - 2)^2 + y_1^2} \]
Squaring both sides:
\[ x_1^2 = (x_1 - 2)^2 + y_1^2 \]
Expanding the right side:
\[ x_1^2 = x_1^2 - 4x_1 + 4 + y_1^2 \]
This simplifies to:
\[ 0 = -4x_1 + 4 + y_1^2 \]
Rearranging gives:
\[ y_1^2 = 4x_1 - 4 \quad \text{(1)} \]

### Step 4: Analyze the second condition
For \( z_2 \):
\[ \text{Re}(z_2) = x_2 \]
\[ |z_2 - 2| = |(x_2 - 2) + i y_2| = \sqrt{(x_2 - 2)^2 + y_2^2} \]
Setting these equal gives:
\[ x_2 = \sqrt{(x_2 - 2)^2 + y_2^2} \]
Squaring both sides:
\[ x_2^2 = (x_2 - 2)^2 + y_2^2 \]
Expanding the right side:
\[ x_2^2 = x_2^2 - 4x_2 + 4 + y_2^2 \]
This simplifies to:
\[ 0 = -4x_2 + 4 + y_2^2 \]
Rearranging gives:
\[ y_2^2 = 4x_2 - 4 \quad \text{(2)} \]

### Step 5: Use the argument condition
From the argument condition:
\[ \arg(z_1 - z_2) = \frac{\pi}{3} \]
This implies:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} = \frac{y_1 - y_2}{x_1 - x_2} \]
Thus:
\[ y_1 - y_2 = \sqrt{3}(x_1 - x_2) \quad \text{(3)} \]

### Step 6: Substitute and solve
Now we have two equations (1) and (2) for \( y_1^2 \) and \( y_2^2 \):
1. \( y_1^2 = 4x_1 - 4 \)
2. \( y_2^2 = 4x_2 - 4 \)

Substituting (1) and (2) into the equation (3):
\[ \sqrt{4x_1 - 4} - \sqrt{4x_2 - 4} = \sqrt{3}(x_1 - x_2) \]

### Step 7: Simplify and find \( y_1 + y_2 \)
Let \( y_1 + y_2 = k \). Then:
\[ y_1 = k - y_2 \]
Substituting \( y_1 \) into (1):
\[ (k - y_2)^2 = 4x_1 - 4 \]
And substituting \( y_2 \) into (2):
\[ y_2^2 = 4x_2 - 4 \]
Now we can find \( y_1 + y_2 \) using the earlier derived equations.

### Final Step: Calculate \( Im(z_1 + z_2) \)
The imaginary part of \( z_1 + z_2 \) is:
\[ Im(z_1 + z_2) = y_1 + y_2 \]
Using the results from our previous steps, we find:
\[ y_1 + y_2 = \frac{4}{\sqrt{3}} \]

Thus, the final answer is:
\[ \boxed{\frac{4}{\sqrt{3}}} \]