If `|z+barz|+|z-barz|=2`, then z lies on

Options

A. a straight line
B. a square
C. a circle
D. none of these

Explanation

To solve the problem, we need to analyze the given equation involving the complex number \( z \) and its conjugate \( \bar{z} \).

### Step-by-Step Solution:

1. **Define the Complex Number**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit.

2. **Find the Conjugate**:
The conjugate of \( z \) is given by \( \bar{z} = x - iy \).

3. **Substitute into the Given Equation**:
The equation provided is:
\[ |z + \bar{z}| + |z - \bar{z}| = 2 \]
Substitute \( z \) and \( \bar{z} \):
\[ |(x + iy) + (x - iy)| + |(x + iy) - (x - iy)| = 2 \]

4. **Simplify the Expressions**:
- For \( z + \bar{z} \):
\[ z + \bar{z} = (x + iy) + (x - iy) = 2x \]
- For \( z - \bar{z} \):
\[ z - \bar{z} = (x + iy) - (x - iy) = 2iy \]

5. **Calculate the Moduli**:
- The modulus of \( z + \bar{z} \):
\[ |z + \bar{z}| = |2x| = 2|x| \]
- The modulus of \( z - \bar{z} \):
\[ |z - \bar{z}| = |2iy| = 2|y| \]

6. **Combine the Results**:
Substitute back into the equation:
\[ 2|x| + 2|y| = 2 \]

7. **Simplify the Equation**:
Divide the entire equation by 2:
\[ |x| + |y| = 1 \]

8. **Interpret the Result**:
The equation \( |x| + |y| = 1 \) represents a geometric figure in the coordinate plane. Specifically, this is the equation of a square (or diamond shape) centered at the origin with vertices at \( (1, 0) \), \( (0, 1) \), \( (-1, 0) \), and \( (0, -1) \).

### Conclusion:
Thus, the complex number \( z \) lies on the boundary of a square (or diamond shape) in the complex plane.