The two lines of regression are parallel to the axes if r = ……………..

Answer

To determine when the two lines of regression are parallel to the axes, we need to analyze the correlation coefficient \( r \).

### Step-by-Step Solution:

1. **Understanding Regression Lines**:
- The two lines of regression are the regression line of \( Y \) on \( X \) and the regression line of \( X \) on \( Y \).
- These lines can be represented as \( Y = a + bX \) and \( X = c + dY \), where \( a, b, c, d \) are constants.

2. **Condition for Parallel Lines**:
- For the regression lines to be parallel to the axes, one line must be horizontal (parallel to the X-axis) and the other must be vertical (parallel to the Y-axis).
- This means that one regression line does not change with the changes in the other variable.

3. **Understanding Correlation Coefficient \( r \)**:
- The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables.
- If \( r = 1 \) or \( r = -1 \), it indicates a perfect linear relationship, while \( r = 0 \) indicates no linear relationship.

4. **Condition for Perpendicularity**:
- The two regression lines are perpendicular to each other when the variables are independent.
- This independence occurs when \( r = 0 \).

5. **Conclusion**:
- Therefore, the two lines of regression are parallel to the axes if \( r = 0 \).

### Final Answer:
The two lines of regression are parallel to the axes if \( r = 0 \).