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 problem \( |z + \bar{z}| + |z - \bar{z}| = 8 \), we will express \( z \) in terms of its real and imaginary parts and analyze the equation step by step.

### Step-by-Step Solution:

1. **Express \( z \) in terms of real and imaginary parts**:
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of \( z \). The conjugate of \( z \) is \( \bar{z} = x - iy \).

2. **Calculate \( z + \bar{z} \) and \( z - \bar{z} \)**:
\[ z + \bar{z} = (x + iy) + (x - iy) = 2x \]
\[ z - \bar{z} = (x + iy) - (x - iy) = 2iy \]

3. **Find the moduli**:
\[ |z + \bar{z}| = |2x| = 2|x| \]
\[ |z - \bar{z}| = |2iy| = 2|y| \]

4. **Substitute into the original equation**:
The equation \( |z + \bar{z}| + |z - \bar{z}| = 8 \) becomes:
\[ 2|x| + 2|y| = 8 \]

5. **Simplify the equation**:
Dividing the entire equation by 2 gives:
\[ |x| + |y| = 4 \]

6. **Interpret the result**:
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 complex number \( z \) lies on the boundary of the diamond shape defined by the equation \( |x| + |y| = 4 \).

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