Let z and w be non - zero complex numbers such that `zw=|z^(2)|` and `|z-barz|+|w+barw|=4.` If w varies, then the perimeter of the locus of z is

Options

A. `8sqrt2` units
B. `4sqrt2` units
C. 8 units
D. 4 units

Answer

To solve the problem, we need to analyze the given conditions step by step.

### Step 1: Understand the given equations
We have two complex numbers \( z \) and \( w \) such that:
1. \( zw = |z|^2 \)
2. \( |z - \bar{z}| + |w + \bar{w}| = 4 \)

### Step 2: Simplify the first equation
From the equation \( zw = |z|^2 \), we can express \( w \) in terms of \( z \):
\[ w = \frac{|z|^2}{z} \]
Since \( z \) is non-zero, this is valid.

### Step 3: Rewrite the second equation
Now, let's analyze the second equation:
\[ |z - \bar{z}| + |w + \bar{w}| = 4 \]
The term \( |z - \bar{z}| \) can be simplified. If we let \( z = x + iy \), then:
\[ z - \bar{z} = (x + iy) - (x - iy) = 2iy \]
Thus,
\[ |z - \bar{z}| = |2iy| = 2|y| \]

### Step 4: Substitute \( w \)
Next, we substitute \( w \):
\[ w + \bar{w} = \frac{|z|^2}{z} + \frac{|z|^2}{\bar{z}} = |z|^2 \left( \frac{1}{z} + \frac{1}{\bar{z}} \right) = |z|^2 \left( \frac{\bar{z} + z}{|z|^2} \right) = \bar{z} + z = 2x \]
Thus,
\[ |w + \bar{w}| = |2x| = 2|x| \]

### Step 5: Combine the results
Now we can rewrite the second equation:
\[ 2|y| + 2|x| = 4 \]
Dividing through by 2 gives:
\[ |x| + |y| = 2 \]

### Step 6: Identify the locus of \( z \)
The equation \( |x| + |y| = 2 \) represents a diamond (or rhombus) shape in the coordinate plane with vertices at \( (2,0) \), \( (0,2) \), \( (-2,0) \), and \( (0,-2) \).

### Step 7: Calculate the perimeter of the locus
The perimeter of a rhombus can be calculated as:
\[ \text{Perimeter} = 4 \times \text{side length} \]
The distance between opposite vertices (from \( (2,0) \) to \( (-2,0) \) or from \( (0,2) \) to \( (0,-2) \)) is 4. The side length can be calculated using the distance formula between two adjacent vertices, for example, from \( (2,0) \) to \( (0,2) \):
\[ \text{side length} = \sqrt{(2 - 0)^2 + (0 - 2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
Thus, the perimeter is:
\[ \text{Perimeter} = 4 \times 2\sqrt{2} = 8\sqrt{2} \]

### Final Answer
The perimeter of the locus of \( z \) is \( 8\sqrt{2} \) units.