If `omega` is any complex number such that `z omega=|z|^(2)` and `|z-barz|+|omega+baromega|=4`, then as `omega` varies, then the area bounded by the locus of `z` is

Options

A. `4` sq. units
B. `8` sq. units
C. `16` sq. units
D. `12` sq. units

Explanation

To solve the problem step by step, we will analyze the given equations and derive the area bounded by the locus of \( z \).

### Step 1: Given Equations
We start with the equations:
1. \( z \cdot \omega = |z|^2 \)
2. \( |z - \bar{z}| + |\omega + \bar{\omega}| = 4 \)

### Step 2: Express \( \omega \)
From the first equation, we can express \( \omega \):
\[ \omega = \frac{|z|^2}{z} \]
Since \( |z|^2 = z \cdot \bar{z} \), we can substitute:
\[ \omega = \frac{z \cdot \bar{z}}{z} = \bar{z} \]

### Step 3: Substitute \( \omega \) in the Second Equation
Now substituting \( \omega = \bar{z} \) into the second equation:
\[ |z - \bar{z}| + |\bar{z} + z| = 4 \]
We know \( |z - \bar{z}| = |2i y| = 2|y| \) (where \( z = x + iy \) and \( \bar{z} = x - iy \)), and \( |\bar{z} + z| = |2x| = 2|x| \). Thus, we have:
\[ 2|y| + 2|x| = 4 \]
Dividing by 2 gives:
\[ |x| + |y| = 2 \]

### Step 4: Identify the Locus
The equation \( |x| + |y| = 2 \) represents a diamond (or rhombus) shape in the coordinate plane with vertices at \( (2, 0), (0, 2), (-2, 0), (0, -2) \).

### Step 5: Calculate the Area
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. The diagonals of our rhombus are both equal to 4 (the distance from \( (2, 0) \) to \( (-2, 0) \) and from \( (0, 2) \) to \( (0, -2) \)):
\[ d_1 = 4, \quad d_2 = 4 \]
Thus, the area is:
\[ A = \frac{1}{2} \times 4 \times 4 = 8 \]

### Conclusion
The area bounded by the locus of \( z \) as \( \omega \) varies is \( 8 \) square units.

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