If a complex number lies in third quadrant ,then its conjugate lies in the ___ quadrant.

Answer

To determine the quadrant in which the conjugate of a complex number lies, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Quadrants**:
- The complex number is represented as \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.
- The quadrants are defined as follows:
- 1st Quadrant: \( x > 0 \), \( y > 0 \)
- 2nd Quadrant: \( x < 0 \), \( y > 0 \)
- 3rd Quadrant: \( x < 0 \), \( y < 0 \)
- 4th Quadrant: \( x > 0 \), \( y < 0 \)

2. **Identifying the Complex Number in the 3rd Quadrant**:
- For a complex number in the 3rd quadrant, both \( x < 0 \) and \( y < 0 \).

3. **Finding the Conjugate**:
- The conjugate of the complex number \( z = x + iy \) is given by \( \overline{z} = x - iy \).
- Here, the real part remains \( x \) and the imaginary part becomes \( -y \).

4. **Analyzing the Conjugate**:
- Since \( x < 0 \) (remains negative) and \( -y > 0 \) (because \( y < 0 \), negating it makes it positive), we have:
- Real part: \( x < 0 \)
- Imaginary part: \( -y > 0 \)

5. **Determining the Quadrant of the Conjugate**:
- The conditions \( x < 0 \) and \( -y > 0 \) indicate that the conjugate lies in the 2nd quadrant.

### Final Answer:
The conjugate of a complex number that lies in the 3rd quadrant lies in the **2nd quadrant**.