If `|z+barz|+|z-barz|=8`, then z lies on

Options

A. a circle
B. a straight line
C. a square
D. an ellipse

Explanation

To solve the equation \( |z + \bar{z}| + |z - \bar{z}| = 8 \), we will follow these steps:

### Step 1: Represent \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of \( z \).

### Step 2: Calculate \( \bar{z} \)
The conjugate of \( z \) is given by:
\[ \bar{z} = x - iy \]

### Step 3: Compute \( z + \bar{z} \) and \( z - \bar{z} \)
Now, we can find:
\[ z + \bar{z} = (x + iy) + (x - iy) = 2x \]
\[ z - \bar{z} = (x + iy) - (x - iy) = 2iy \]

### Step 4: Find the moduli
Next, we calculate the moduli:
\[ |z + \bar{z}| = |2x| = 2|x| \]
\[ |z - \bar{z}| = |2iy| = 2|y| \]

### Step 5: Substitute into the original equation
Substituting these moduli into the equation gives:
\[ 2|x| + 2|y| = 8 \]

### Step 6: Simplify the equation
Dividing the entire equation by 2, we get:
\[ |x| + |y| = 4 \]

### Step 7: Interpret the equation geometrically
The equation \( |x| + |y| = 4 \) represents a diamond (or rhombus) shape in the coordinate plane, with vertices at \( (4, 0) \), \( (0, 4) \), \( (-4, 0) \), and \( (0, -4) \).

### Conclusion
Thus, the set of points \( z \) that satisfy the given equation lies on the boundary of the diamond defined by \( |x| + |y| = 4 \).

---