If `|z+barz|=|z-barz|`, then value of locus of z is

Options

A. a pair of straight line
B. a rectangular hyperbola
C. a line
D. a set of four lines

Explanation

To solve the problem, we need to analyze the equation given and derive the locus of the complex number \( z \).

### Step-by-Step Solution:

1. **Express \( z \) 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 \( z \).

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

3. **Set up the equation**:
We are given that \( |z + \bar{z}| = |z - \bar{z}| \).

4. **Calculate \( z + \bar{z} \)**:
\[ z + \bar{z} = (x + iy) + (x - iy) = 2x \]
Therefore, \( |z + \bar{z}| = |2x| = 2|x| \).

5. **Calculate \( z - \bar{z} \)**:
\[ z - \bar{z} = (x + iy) - (x - iy) = 2iy \]
Therefore, \( |z - \bar{z}| = |2iy| = 2|y| \).

6. **Set the two moduli equal**:
From the given condition, we have:
\[ 2|x| = 2|y| \]
Dividing both sides by 2 gives:
\[ |x| = |y| \]

7. **Interpret the result**:
The equation \( |x| = |y| \) represents two lines in the Cartesian plane:
- \( x = y \) (the line at 45 degrees)
- \( x = -y \) (the line at -45 degrees)

8. **Conclusion**:
The locus of \( z \) is a pair of straight lines given by the equations \( x + y = 0 \) and \( x - y = 0 \).

### Final Answer:
The locus of \( z \) is a pair of straight lines represented by the equations \( x + y = 0 \) and \( x - y = 0 \).

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