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 start with the equation given:

\[ |z + \bar{z}| + |z - \bar{z}| = 2 \]

### Step 1: Express \( z \) in terms of \( x \) and \( y \)

Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The conjugate of \( z \) is given by:

\[ \bar{z} = x - iy \]

### Step 2: Calculate \( z + \bar{z} \) and \( z - \bar{z} \)

Now, we can compute:

\[ z + \bar{z} = (x + iy) + (x - iy) = 2x \]

\[ z - \bar{z} = (x + iy) - (x - iy) = 2iy \]

### Step 3: Substitute into the modulus equation

Substituting these into the original equation, we have:

\[ |2x| + |2iy| = 2 \]

### Step 4: Simplify the equation

Since the modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \), we can simplify:

\[ |2x| + |2y| = 2 \]

Dividing both sides by 2 gives:

\[ |x| + |y| = 1 \]

### Step 5: Interpret the equation geometrically

The equation \( |x| + |y| = 1 \) represents a geometric figure in the coordinate plane. This is the equation of a diamond (or square rotated 45 degrees) 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 in the complex plane.