If two lines of Regression are respectively y=ax+b and x=`alphay+beta.` If the two variables have the same mean, then `b//beta`=

Options

A. `(1+a)/(1+alpha)`
B. `(1+alpha)/(1+a)`
C. `(1-a)/(1-alpha)`
D. `(1-alpha)/(1-a)`

Answer

To solve the problem, we need to analyze the given regression equations and the condition that the means of the two variables are the same.

### Step-by-Step Solution:

1. **Understand the Regression Lines**:
We have two regression lines:
- \( y = ax + b \) (Equation 1)
- \( x = \alpha y + \beta \) (Equation 2)

2. **Condition of Means**:
We are given that the means of the two variables are the same, which implies:
\[ \bar{x} = \bar{y} \]

3. **Setting the Intersection Point**:
The intersection point of the two regression lines corresponds to the means of \( x \) and \( y \). Since \( \bar{x} = \bar{y} \), we can denote this common mean as \( m \). Therefore, we can set:
\[ x = m \quad \text{and} \quad y = m \]

4. **Substituting into the Equations**:
Substitute \( x = m \) into Equation 1:
\[ m = am + b \]
Rearranging gives:
\[ m(1 - a) = b \quad \text{(Equation 3)} \]

Now substitute \( y = m \) into Equation 2:
\[ m = \alpha m + \beta \]
Rearranging gives:
\[ m(1 - \alpha) = \beta \quad \text{(Equation 4)} \]

5. **Finding the Ratio \( \frac{b}{\beta} \)**:
From Equations 3 and 4, we can express \( b \) and \( \beta \):
\[ b = m(1 - a) \quad \text{and} \quad \beta = m(1 - \alpha) \]
Now, we can find the ratio:
\[ \frac{b}{\beta} = \frac{m(1 - a)}{m(1 - \alpha)} \]
Since \( m \) is common in both the numerator and denominator, it cancels out:
\[ \frac{b}{\beta} = \frac{1 - a}{1 - \alpha} \]

6. **Final Answer**:
Therefore, the final result is:
\[ \frac{b}{\beta} = \frac{1 - a}{1 - \alpha} \]