If the regression equation of y on x is `-2x + 5y= 14` and the regression equation of x on y is given by `mx-y + 10= 0`. If the coefficient of correlation between x and y is `(1)/(sqrt10)`, then the value of m is

Options

A. 6
B. 4
C. 8
D. 2

Answer

To solve the problem, we need to find the value of \( m \) given the regression equations and the correlation coefficient. Let's break it down step by step.

### Step 1: Rewrite the regression equation of \( y \) on \( x \)

The regression equation of \( y \) on \( x \) is given as:
\[ -2x + 5y = 14 \]
We can rearrange this to express \( y \) in terms of \( x \):
\[ 5y = 2x + 14 \]
\[ y = \frac{2}{5}x + \frac{14}{5} \]
From this, we can identify the slope \( b_{xy} \) as:
\[ b_{xy} = \frac{2}{5} \]

### Step 2: Rewrite the regression equation of \( x \) on \( y \)

The regression equation of \( x \) on \( y \) is given as:
\[ mx - y + 10 = 0 \]
Rearranging gives:
\[ mx = y - 10 \]
\[ x = \frac{1}{m}y - \frac{10}{m} \]
From this, we can identify the slope \( b_{yx} \) as:
\[ b_{yx} = \frac{1}{m} \]

### Step 3: Use the correlation coefficient

We know the coefficient of correlation \( r \) is given as:
\[ r = \frac{1}{\sqrt{10}} \]
The relationship between the slopes and the correlation coefficient is given by:
\[ r = \sqrt{b_{xy} \cdot b_{yx}} \]
Substituting the values we have:
\[ \frac{1}{\sqrt{10}} = \sqrt{\left(\frac{2}{5}\right) \cdot \left(\frac{1}{m}\right)} \]

### Step 4: Square both sides to eliminate the square root

Squaring both sides gives:
\[ \frac{1}{10} = \left(\frac{2}{5}\right) \cdot \left(\frac{1}{m}\right) \]

### Step 5: Solve for \( m \)

Rearranging the equation:
\[ \frac{1}{10} = \frac{2}{5m} \]
Cross-multiplying gives:
\[ 1 \cdot 5m = 2 \cdot 10 \]
\[ 5m = 20 \]
Dividing both sides by 5:
\[ m = 4 \]

### Final Answer
Thus, the value of \( m \) is:
\[ \boxed{4} \]