Lines of regression are ………….. If `r=0` and they are ………….. If `r=pm1`.

Answer

To solve the question regarding the nature of lines of regression based on the correlation coefficient \( r \), we can break it down into two parts:

### Step-by-Step Solution:

1. **Understanding the Correlation Coefficient \( r \)**:
- The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1.

2. **Case 1: When \( r = 0 \)**:
- When \( r = 0 \), it indicates that there is no linear correlation between the two variables. This means that the regression lines do not have a predictable relationship.
- In this case, the two lines of regression are **perpendicular** to each other.

3. **Case 2: When \( r = +1 \) or \( r = -1 \)**:
- When \( r = +1\), it indicates a perfect positive linear correlation, meaning that as one variable increases, the other variable also increases in a perfectly linear manner.
- When \( r = -1\), it indicates a perfect negative linear correlation, meaning that as one variable increases, the other variable decreases in a perfectly linear manner.
- In both cases (whether \( r = +1 \) or \( r = -1 \)), the two lines of regression are **the same**.

### Final Answer:
- Lines of regression are **perpendicular** if \( r = 0 \) and they are **the same** if \( r = +1 \) or \( r = -1 \).