If the lines of regression of `y` on `x` is `y=x/4` and `x` on `y` is `x=y/9 +1`, then the value of `r` is

Options

A. `1/6`
B. `0`
C. `-1/4`
D. `-1/6`

Answer

To find the value of the correlation coefficient \( r \) given the lines of regression, we can follow these steps:

### Step 1: Identify the regression equations
We are given two regression equations:
1. The regression of \( y \) on \( x \): \( y = \frac{x}{4} \)
2. The regression of \( x \) on \( y \): \( x = \frac{y}{9} + 1 \)

### Step 2: Extract the regression coefficients
From the regression line of \( y \) on \( x \):
- The slope \( b_{yx} \) (regression coefficient of \( y \) on \( x \)) is \( \frac{1}{4} \).

From the regression line of \( x \) on \( y \):
- The slope \( b_{xy} \) (regression coefficient of \( x \) on \( y \)) is \( \frac{1}{9} \).

### Step 3: Use the formula for the correlation coefficient
The formula for the correlation coefficient \( r \) in terms of the regression coefficients is:
\[ r^2 = b_{xy} \cdot b_{yx} \]
Substituting the values we found:
\[ r^2 = \left(\frac{1}{9}\right) \cdot \left(\frac{1}{4}\right) = \frac{1}{36} \]

### Step 4: Calculate \( r \)
To find \( r \), we take the square root of \( r^2 \):
\[ r = \sqrt{\frac{1}{36}} = \frac{1}{6} \]

Since both regression coefficients are positive, \( r \) will also be positive.

### Final Answer
Thus, the value of \( r \) is:
\[ \boxed{\frac{1}{6}} \]