The regression lines of x on y and y on x are `x=4y+5 and y=kx+4` respectively. The limits of k are given by

Options

A. `0ltklt(1)/(4)`
B. `-1ltklt(1)/(2)`
C. `-1ltklt1`
D. none of these

Answer

To solve the problem, we need to find the limits of \( k \) given the regression lines \( x = 4y + 5 \) and \( y = kx + 4 \).

### Step-by-Step Solution:

1. **Identify the Regression Coefficients:**
- The regression line of \( x \) on \( y \) is given by \( x = 4y + 5 \). Here, the coefficient of \( y \) is \( b_{xy} = 4 \).
- The regression line of \( y \) on \( x \) is given by \( y = kx + 4 \). Here, the coefficient of \( x \) is \( b_{yx} = k \).

2. **Relate the Regression Coefficients to the Correlation Coefficient:**
- The correlation coefficient \( r \) can be expressed in terms of the regression coefficients as:
\[ r = \sqrt{b_{xy} \cdot b_{yx}} = \sqrt{4k} \]

3. **Determine the Range of the Correlation Coefficient:**
- The correlation coefficient \( r \) must satisfy the condition \( -1 \leq r \leq 1 \).
- Since both regression coefficients \( b_{xy} \) and \( b_{yx} \) are positive, \( r \) will also be positive. Thus, we only need to consider the range \( 0 \leq r \leq 1 \).

4. **Set Up the Inequality:**
- From the expression for \( r \):
\[ 0 \leq \sqrt{4k} \leq 1 \]
- Squaring all parts of the inequality (since \( r \) is non-negative):
\[ 0 \leq 4k \leq 1 \]

5. **Solve for \( k \):**
- Dividing the entire inequality by 4 gives:
\[ 0 \leq k \leq \frac{1}{4} \]

### Conclusion:
The limits of \( k \) are:
\[ 0 \leq k \leq \frac{1}{4} \]