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)/(sqrt3)`
D. `-(1)/(sqrt3)`

Explanation

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 of the regression lines**:
- 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. **Calculate the slopes using tangent**:
- The slope \( b_{YX} \) (slope of \( Y \) on \( X \)) is given by:
\[ b_{YX} = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
- The slope \( b_{XY} \) (slope of \( X \) on \( Y \)) is given by:
\[ b_{XY} = \tan(60^\circ) = \sqrt{3} \]

3. **Use the relationship between slopes and correlation coefficient**:
- The relationship between the slopes and the correlation coefficient \( r \) is given by:
\[ r = \sqrt{b_{XY} \cdot b_{YX}} \]

4. **Substitute the values of the slopes**:
- Substitute \( b_{YX} \) and \( b_{XY} \) into the formula:
\[ r = \sqrt{\left(\sqrt{3}\right) \cdot \left(\frac{1}{\sqrt{3}}\right)} \]

5. **Simplify the expression**:
- Simplifying the expression gives:
\[ r = \sqrt{1} = 1 \]

6. **Determine the sign of \( r \)**:
- Since both slopes \( b_{YX} \) and \( b_{XY} \) are positive, the correlation coefficient \( r \) will also be positive.

7. **Final answer**:
- Therefore, the value of \( r(X, Y) \) is:
\[ r(X, Y) = 1 \]

### Conclusion:
The correlation coefficient \( r(X, Y) \) is \( 1 \).