If `|z_(1)| = 2 and (1-i)z_(2) + (1+i)bar(z)_(2) = 8 sqrt(2)`, then

Options

A. minimum value of `|z_(1) - z_(2)|` is 1
B. minimum value of `|z_(1) - z_(2)|` is 2
C. maximum value of `|z_(1) - z_(2)|` is 8
D. maximum value of `|z_(1) - z_(2)|` is 4

Answer

To solve the problem step by step, we need to analyze the given conditions and derive the required values.

### Step 1: Understand the Given Conditions
We have two complex numbers, \( z_1 \) and \( z_2 \). The modulus of \( z_1 \) is given as \( |z_1| = 2 \). This means that \( z_1 \) lies on a circle of radius 2 centered at the origin in the complex plane.

### Step 2: Rewrite the Equation
The equation given is:
\[ (1 - i)z_2 + (1 + i)\overline{z_2} = 8\sqrt{2} \]
Let’s express \( z_2 \) in terms of its real and imaginary parts:
\[ z_2 = x_2 + iy_2 \]
Then, the conjugate \( \overline{z_2} \) is:
\[ \overline{z_2} = x_2 - iy_2 \]
Substituting these into the equation gives:
\[ (1 - i)(x_2 + iy_2) + (1 + i)(x_2 - iy_2) = 8\sqrt{2} \]

### Step 3: Expand the Equation
Expanding both parts:
\[ (1 - i)(x_2 + iy_2) = x_2 + iy_2 - ix_2 - y_2 = (x_2 + y_2) + i(y_2 - x_2) \]
\[ (1 + i)(x_2 - iy_2) = x_2 - iy_2 + ix_2 + y_2 = (x_2 + y_2) + i(x_2 - y_2) \]
Adding these results:
\[ 2(x_2 + y_2) + i(y_2 - x_2 + x_2 - y_2) = 2(x_2 + y_2) \]
Setting this equal to \( 8\sqrt{2} \):
\[ 2(x_2 + y_2) = 8\sqrt{2} \]
Dividing both sides by 2:
\[ x_2 + y_2 = 4\sqrt{2} \]

### Step 4: Interpret the Result
The equation \( x_2 + y_2 = 4\sqrt{2} \) represents a line in the complex plane.

### Step 5: Find the Minimum and Maximum Values of \( |z_1 - z_2| \)
To find the minimum and maximum values of \( |z_1 - z_2| \), we need to consider the distance from the center of the circle (the origin) to the line \( x_2 + y_2 = 4\sqrt{2} \).

### Step 6: Calculate the Distance from the Origin to the Line
The line can be rewritten in the standard form:
\[ x + y - 4\sqrt{2} = 0 \]
Using the distance formula from a point to a line \( Ax + By + C = 0 \):
\[ \text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1, B = 1, C = -4\sqrt{2}, (x_0, y_0) = (0, 0) \):
\[ \text{Distance} = \frac{|0 + 0 - 4\sqrt{2}|}{\sqrt{1^2 + 1^2}} = \frac{4\sqrt{2}}{\sqrt{2}} = 4 \]

### Step 7: Determine Minimum and Maximum Distances
- The minimum distance from the origin to the line is \( 4 \).
- Since \( |z_1| = 2 \), the minimum value of \( |z_1 - z_2| \) is:
\[ \text{Minimum} = \text{Distance to line} - \text{Radius} = 4 - 2 = 2 \]
- The maximum value of \( |z_1 - z_2| \) is:
\[ \text{Maximum} = \text{Distance to line} + \text{Radius} = 4 + 2 = 6 \]

### Conclusion
Thus, the minimum value of \( |z_1 - z_2| \) is 2 and the maximum value is 6.