If the slopes of the line of regression of Y and X and of X and Y are `30^(@)` and `60^(@)` respectively, then r (X,Y) is

Options

A. -1
B. 1
C. `(1)/sqrt(3)`
D. `-(1)/sqrt(3)`

Answer

To solve the problem, we need to find the correlation coefficient \( r(X, Y) \) given the slopes of the regression lines of \( Y \) on \( X \) and \( X \) on \( Y \).

### Step-by-Step Solution:

1. **Identify the Slopes**:
- The slope of the regression line of \( Y \) on \( X \) is given as \( 30^\circ \).
- The slope of the regression line of \( X \) on \( Y \) is given as \( 60^\circ \).

2. **Convert Angles to Tangent Values**:
- The slope of the regression line of \( Y \) on \( X \) can be represented as \( B_{Y|X} = \tan(30^\circ) \).
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).
- The slope of the regression line of \( X \) on \( Y \) can be represented as \( B_{X|Y} = \tan(60^\circ) \).
- We know that \( \tan(60^\circ) = \sqrt{3} \).

3. **Relate the Slopes to the Correlation Coefficient**:
- The relationship between the slopes and the correlation coefficient \( r \) is given by:
\[ B_{Y|X} \cdot B_{X|Y} = r^2 \]
- Substituting the values we found:
\[ \left(\frac{1}{\sqrt{3}}\right) \cdot \left(\frac{1}{\sqrt{3}}\right) = r^2 \]
- This simplifies to:
\[ \frac{1}{3} = r^2 \]

4. **Calculate the Correlation Coefficient**:
- To find \( r \), we take the square root of both sides:
\[ r = \sqrt{\frac{1}{3}} \text{ or } r = -\sqrt{\frac{1}{3}} \]
- Thus, we can express \( r \) as:
\[ r = \frac{1}{\sqrt{3}} \text{ or } r = -\frac{1}{\sqrt{3}} \]

5. **Choose the Correct Option**:
- Since the correlation coefficient can be either positive or negative depending on the context, but the options provided are:
- Option A: -1
- Option B: 1
- Option C: \( \frac{1}{\sqrt{3}} \)
- Option D: \( -\frac{1}{\sqrt{3}} \)
- The correct answer based on our calculation is \( -\frac{1}{\sqrt{3}} \) (Option D).

### Final Answer:
Thus, \( r(X, Y) = -\frac{1}{\sqrt{3}} \).