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 given equation \( |z + \bar{z}| = |z - \bar{z}| \).

### Step 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.

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

### Step 3: Substitute \( z \) and \( \bar{z} \) into the equation
Now, we can rewrite the left-hand side and right-hand side of the given equation:
\[ z + \bar{z} = (x + iy) + (x - iy) = 2x \]
\[ z - \bar{z} = (x + iy) - (x - iy) = 2iy \]

### Step 4: Apply the modulus
Now, we take the modulus of both sides:
\[ |z + \bar{z}| = |2x| = 2|x| \]
\[ |z - \bar{z}| = |2iy| = 2|y| \]

### Step 5: Set the moduli equal to each other
According to the problem, we have:
\[ 2|x| = 2|y| \]

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

### Step 7: Interpret the result
The equation \( |x| = |y| \) represents two lines in the Cartesian plane:
1. \( y = x \)
2. \( y = -x \)

### Conclusion
Thus, the locus of \( z \) is the union of the lines \( y = x \) and \( y = -x \).

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