If `z_(1)` and `z_(2)` satisfy the equation `|z-2|=|"Re"(z)|` and arg`(z1-z2)=pi/3, then Im (z1+z2) =k/sqrt 3 where k is

Options

A. 0
B. `+-pi/2`
C. `+-pi`
D. `+-pi/4`

Answer

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

### Step 1: Understand the given equation
We have the equation:
\[ |z - 2| = |\text{Re}(z)| \]
Let \( z = x + iy \), where \( x = \text{Re}(z) \) and \( y = \text{Im}(z) \). Thus, we can rewrite the equation as:
\[ |(x - 2) + iy| = |x| \]

### Step 2: Express the modulus
The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). Therefore:
\[ \sqrt{(x - 2)^2 + y^2} = |x| \]

### Step 3: Square both sides
Squaring both sides gives:
\[ (x - 2)^2 + y^2 = x^2 \]

### Step 4: Expand and simplify
Expanding the left side:
\[ x^2 - 4x + 4 + y^2 = x^2 \]
Now, subtract \( x^2 \) from both sides:
\[ -4x + 4 + y^2 = 0 \]
Rearranging gives:
\[ y^2 = 4x - 4 \]
This can be simplified to:
\[ y^2 = 4(x - 1) \]
This is our first equation.

### Step 5: Set up equations for \( z_1 \) and \( z_2 \)
Let \( z_1 = x_1 + iy_1 \) and \( z_2 = x_2 + iy_2 \). From the first equation, we have:
1. \( y_1^2 = 4(x_1 - 1) \)
2. \( y_2^2 = 4(x_2 - 1) \)

### Step 6: Use the argument condition
We are given that:
\[ \arg(z_1 - z_2) = \frac{\pi}{3} \]
This implies:
\[ \frac{y_1 - y_2}{x_1 - x_2} = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Thus, we can write:
\[ y_1 - y_2 = \sqrt{3}(x_1 - x_2) \]
This is our second equation.

### Step 7: Solve the equations
Now we have two equations:
1. \( y_1^2 - y_2^2 = 4(x_1 - x_2) \)
2. \( y_1 - y_2 = \sqrt{3}(x_1 - x_2) \)

Using the identity \( a^2 - b^2 = (a - b)(a + b) \):
\[ (y_1 - y_2)(y_1 + y_2) = 4(x_1 - x_2) \]
Substituting \( y_1 - y_2 \) from the second equation:
\[ \sqrt{3}(x_1 - x_2)(y_1 + y_2) = 4(x_1 - x_2) \]

### Step 8: Simplify
Assuming \( x_1 \neq x_2 \), we can divide both sides by \( x_1 - x_2 \):
\[ \sqrt{3}(y_1 + y_2) = 4 \]
Thus:
\[ y_1 + y_2 = \frac{4}{\sqrt{3}} \]

### Step 9: Find \( k \)
From the problem statement, we have:
\[ \text{Im}(z_1 + z_2) = \frac{k}{\sqrt{3}} \]
Comparing gives:
\[ \frac{4}{\sqrt{3}} = \frac{k}{\sqrt{3}} \]
Thus, \( k = 4 \).

### Final Answer
The value of \( k \) is:
\[ \boxed{4} \]