Let `z and omega` be complex numbers. If `Re(z) = |z-2|, Re(omega) = |omega - 2| and arg(z - omega) = pi/3`, then the value of `Im(z+w)`, is

Options

A. `(1)/sqrt(3)`
B. `(2)/sqrt(3)`
C. `sqrt(3)`
D. `(4)/(sqrt(3))`

Answer

To solve the problem step by step, we will analyze the conditions given and derive the necessary equations.

### Step 1: Define the Complex Numbers
Let \( z = x + iy \) and \( w = u + iv \), where \( x, y, u, v \) are real numbers.

### Step 2: Use the Given Conditions
We have two conditions:
1. \( \text{Re}(z) = |z - 2| \)
2. \( \text{Re}(w) = |w - 2| \)

### Step 3: Express the First Condition
From the first condition:
\[ \text{Re}(z) = x = |z - 2| = |(x - 2) + iy| = \sqrt{(x - 2)^2 + y^2} \]
Thus, we can write:
\[ x = \sqrt{(x - 2)^2 + y^2} \]

### Step 4: Square Both Sides
Squaring both sides gives:
\[ x^2 = (x - 2)^2 + y^2 \]
Expanding the right-hand side:
\[ x^2 = x^2 - 4x + 4 + y^2 \]
This simplifies to:
\[ 0 = -4x + 4 + y^2 \implies y^2 = 4x - 4 \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ y^2 = 4(x - 1) \]
This is the equation of a parabola.

### Step 6: Express the Second Condition
Similarly, for \( w \):
\[ \text{Re}(w) = u = |w - 2| = |(u - 2) + iv| = \sqrt{(u - 2)^2 + v^2} \]
Thus, we can write:
\[ u = \sqrt{(u - 2)^2 + v^2} \]
Squaring both sides gives:
\[ u^2 = (u - 2)^2 + v^2 \]
Expanding and simplifying gives:
\[ v^2 = 4(u - 1) \]

### Step 7: Parametric Form of the Parabolas
From the equations \( y^2 = 4(x - 1) \) and \( v^2 = 4(u - 1) \), we can express \( z \) and \( w \) in parametric form:
- Let \( z = (1 + t_1^2) + 2t_1i \) for some parameter \( t_1 \).
- Let \( w = (1 + t_2^2) + 2t_2i \) for some parameter \( t_2 \).

### Step 8: Use the Argument Condition
Given that \( \arg(z - w) = \frac{\pi}{3} \):
\[ z - w = [(1 + t_1^2) - (1 + t_2^2)] + [2t_1 - 2t_2]i = (t_1^2 - t_2^2) + 2(t_1 - t_2)i \]
The argument can be expressed as:
\[ \arg(z - w) = \tan^{-1}\left(\frac{2(t_1 - t_2)}{t_1^2 - t_2^2}\right) = \frac{\pi}{3} \]
This implies:
\[ \frac{2(t_1 - t_2)}{t_1^2 - t_2^2} = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Step 9: Simplifying the Argument Condition
We can simplify:
\[ \frac{2(t_1 - t_2)}{(t_1 - t_2)(t_1 + t_2)} = \sqrt{3} \]
Assuming \( t_1 \neq t_2 \), we can cancel \( t_1 - t_2 \):
\[ \frac{2}{t_1 + t_2} = \sqrt{3} \implies t_1 + t_2 = \frac{2}{\sqrt{3}} \]

### Step 10: Find the Imaginary Part of \( z + w \)
The imaginary part of \( z + w \) is:
\[ \text{Im}(z + w) = 2t_1 + 2t_2 = 2(t_1 + t_2) = 2 \cdot \frac{2}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \text{Im}(z + w) \) is:
\[ \frac{4}{\sqrt{3}} \]