If ` |z _1 |= 2 and (1 - i)z_2 + (1+i)barz_2 = 8sqrt2`, then the minimum value of ` |z_1 - z_2|` is ______.

Explanation

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the given information
We have two complex numbers \( z_1 \) and \( z_2 \). We know that:
- The modulus of \( z_1 \) is \( |z_1| = 2 \), which means \( z_1 \) lies on a circle centered at the origin with radius 2.
- The equation given is \( (1 - i)z_2 + (1 + i)\overline{z_2} = 8\sqrt{2} \).

### Step 2: Express \( z_2 \) in terms of its real and imaginary parts
Let \( z_2 = x + iy \), where \( x \) and \( y \) are real numbers. The conjugate \( \overline{z_2} \) is \( x - iy \).

### Step 3: Substitute \( z_2 \) and simplify the equation
Substituting \( z_2 \) into the equation:
\[ (1 - i)(x + iy) + (1 + i)(x - iy) = 8\sqrt{2} \]
Expanding this:
\[ (1 - i)(x + iy) = x + iy - ix - y = (x + y) + i(y - x) \]
\[ (1 + i)(x - iy) = x - iy + ix + y = (x + y) + i(x - y) \]
Adding these two results:
\[ (x + y) + i(y - x) + (x + y) + i(x - y) = 2(x + y) + i(0) = 2(x + y) \]
Setting this equal to \( 8\sqrt{2} \):
\[ 2(x + y) = 8\sqrt{2} \]
Dividing both sides by 2:
\[ x + y = 4\sqrt{2} \]

### Step 4: Identify the geometric representation
The equation \( x + y = 4\sqrt{2} \) represents a straight line in the complex plane. The circle \( |z_1| = 2 \) can be represented by the equation \( x^2 + y^2 = 4 \).

### Step 5: Find the minimum distance between the line and the circle
To find the minimum distance between the line \( x + y = 4\sqrt{2} \) and the circle \( x^2 + y^2 = 4 \), we can calculate the perpendicular distance from the center of the circle (0, 0) to the line.

The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( ax + by + c = 0 \) is:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]
For the line \( x + y - 4\sqrt{2} = 0 \), we have:
- \( a = 1 \)
- \( b = 1 \)
- \( c = -4\sqrt{2} \)

Substituting \( (x_1, y_1) = (0, 0) \):
\[ d = \frac{|1 \cdot 0 + 1 \cdot 0 - 4\sqrt{2}|}{\sqrt{1^2 + 1^2}} = \frac{|-4\sqrt{2}|}{\sqrt{2}} = \frac{4\sqrt{2}}{\sqrt{2}} = 4 \]

### Step 6: Calculate the minimum distance
The radius of the circle is 2. Therefore, the minimum distance \( |z_1 - z_2| \) is:
\[ \text{Minimum Distance} = d - \text{radius} = 4 - 2 = 2 \]

### Final Answer
The minimum value of \( |z_1 - z_2| \) is \( \boxed{2} \).

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