If z = `barz` then z lies on

Options

A. (a) x - axis
B. (b) y - axis
C. (c) origin
D. (d) none of these

Answer

To solve the problem where \( z = \bar{z} \), we need to analyze what this means in terms of the complex number \( z \).

### Step-by-Step Solution:

1. **Understand the Definition of \( z \) and \( \bar{z} \)**:
- Let \( z \) be a complex number represented as \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.
- The conjugate of \( z \), denoted as \( \bar{z} \), is given by \( \bar{z} = x - iy \).

2. **Set Up the Equation**:
- According to the problem, we have \( z = \bar{z} \).
- Substituting the definitions, we get:
\[ x + iy = x - iy \]

3. **Equate Real and Imaginary Parts**:
- From the equation \( x + iy = x - iy \), we can separate the real and imaginary parts:
- The real part: \( x = x \) (which is always true).
- The imaginary part: \( iy = -iy \).

4. **Solve for the Imaginary Part**:
- From the imaginary part, we have:
\[ iy + iy = 0 \implies 2iy = 0 \]
- This implies that \( y = 0 \).

5. **Interpret the Result**:
- Since \( y = 0 \), the complex number \( z \) can be expressed as:
\[ z = x + 0i = x \]
- This means that \( z \) is purely real.

6. **Conclusion**:
- A complex number is purely real if its imaginary part is zero, which means it lies on the x-axis of the complex plane.

### Final Answer:
Thus, the complex number \( z \) lies on the **x-axis**.