Which of the following is true?

Options

A. The two lines of regression can never be Parallel
B. The two lines of regression coincide if `e^(2)=1`
C. The two lines of regression ate at right angles if `e=0`
D. If the two variables x and y are related linearly, then `e=-1 or 1`

Answer

To determine which of the statements regarding the angle between two regression lines is true, we will analyze each statement based on the provided information about the angle formula and the properties of regression lines.

### Step-by-Step Solution:

1. **Understanding the Angle Formula**:
The angle θ between two regression lines is given by the formula:
\[ \theta = \tan^{-1} \left( \frac{\sigma_x \cdot \sigma_y}{\sigma_x^2 + \sigma_y^2 \cdot e^2 - 1} \right) \]
where σ_x and σ_y are the standard deviations of variables X and Y, respectively, and e is the correlation coefficient between X and Y.

2. **Analyzing the Case When e = 1**:
- If \( e = 1 \):
\[ \theta = \tan^{-1} \left( \frac{\sigma_x \cdot \sigma_y}{\sigma_x^2 + \sigma_y^2 \cdot 1 - 1} \right) \]
This simplifies to:
\[ \theta = \tan^{-1}(0) = 0 \]
- A θ of 0 means the two regression lines are coincident (they overlap completely).

3. **Analyzing the Case When e = 0**:
- If \( e = 0 \):
\[ \theta = \tan^{-1} \left( \frac{\sigma_x \cdot \sigma_y}{\sigma_x^2 + \sigma_y^2 \cdot 0 - 1} \right) \]
This leads to:
\[ \theta = \tan^{-1}(\infty) = \frac{\pi}{2} \]
- A θ of \( \frac{\pi}{2} \) indicates that the two regression lines are perpendicular.

4. **Analyzing the Case When e = -1 or e = 1**:
- If \( e = 1 \), the correlation is perfectly positive.
- If \( e = -1 \), the correlation is perfectly negative.
- In both cases, the relationship between X and Y is linear.

5. **Conclusion**:
- From the analysis, we can conclude:
- When \( e = 1 \), the lines are coincident.
- When \( e = 0 \), the lines are perpendicular.
- When \( e = -1 \) or \( e = 1 \), the relationship is perfectly linear.

Thus, all four statements regarding the relationships and angles between the regression lines are true.

### Final Answer:
All four statements (A, B, C, D) are true.