In sub-parts (i) and (ii) choose the correct option and in sub-parts (iii) to (v), answer the questions as instructed.
Given that `y=mx+c " and "x=4y+5` are two regression lines y on x and x on y respectively, such that `0 le m le k`. Then k is equals to

Options

A. 1
B. `1/2`
C. 0.25
D. None of these

Answer

To solve the problem, we need to analyze the given regression lines and find the value of \( k \) such that \( 0 \leq m \leq k \).

### Step-by-Step Solution:

1. **Identify the Regression Lines:**
- The first regression line is given by \( y = mx + c \), which represents the regression of \( y \) on \( x \).
- The second regression line is given by \( x = 4y + 5 \), which represents the regression of \( x \) on \( y \).

2. **Extract the Regression Coefficients:**
- From the first regression line \( y = mx + c \), we can identify the regression coefficient \( b_{yx} = m \).
- From the second regression line \( x = 4y + 5 \), we can identify the regression coefficient \( b_{xy} = 4 \).

3. **Use the Relationship Between Regression Coefficients:**
- The product of the regression coefficients is related to the square of the correlation coefficient \( r \):
\[ b_{yx} \cdot b_{xy} = r^2 \]
- Substituting the values we have:
\[ m \cdot 4 = r^2 \]
- Therefore:
\[ r^2 = 4m \]

4. **Determine the Range of \( r^2 \):**
- The correlation coefficient \( r \) lies between 0 and 1, which means:
\[ 0 \leq r^2 \leq 1 \]
- This gives us:
\[ 0 \leq 4m \leq 1 \]

5. **Solve for \( m \):**
- Dividing the entire inequality by 4:
\[ 0 \leq m \leq \frac{1}{4} \]
- This indicates that the maximum value of \( m \) is \( \frac{1}{4} \).

6. **Conclusion:**
- Since \( 0 \leq m \leq k \) and we found \( m \) can go up to \( \frac{1}{4} \), we conclude that:
\[ k = \frac{1}{4} \]

### Final Answer:
\[ k = \frac{1}{4} \]