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 two regression equations given and derive the relationship between \( b \) and \( \beta \) under the condition that the means of the two variables are equal.

### Step-by-Step Solution:

1. **Understanding the Regression Equations**:
We have two regression equations:
\[ y = ax + b \quad \text{(1)} \]
\[ x = \alpha y + \beta \quad \text{(2)} \]

2. **Setting the Means Equal**:
Given that the two variables have the same mean, we denote the means as \( x̄ \) and \( ȳ \). Therefore, we can set:
\[ x̄ = ȳ \]

3. **Substituting Means into the Equations**:
From equation (1), substituting \( x = y = x̄ \):
\[ ȳ = ax̄ + b \]
Since \( x̄ = ȳ \), we can rewrite this as:
\[ x̄ = ax̄ + b \quad \text{(3)} \]

4. **Rearranging Equation (3)**:
Rearranging equation (3) gives:
\[ x̄ - ax̄ = b \]
Factoring out \( x̄ \):
\[ x̄(1 - a) = b \]
Thus, we can express \( x̄ \) as:
\[ x̄ = \frac{b}{1 - a} \quad \text{(4)} \]

5. **Substituting Back into the Second Equation**:
Now, substitute \( ȳ = x̄ \) into equation (2):
\[ x̄ = \alpha ȳ + \beta \]
Replacing \( ȳ \) with \( x̄ \):
\[ x̄ = \alpha x̄ + \beta \]

6. **Rearranging this Equation**:
Rearranging gives:
\[ x̄ - \alpha x̄ = \beta \]
Factoring out \( x̄ \):
\[ x̄(1 - \alpha) = \beta \]
Thus, we can express \( x̄ \) as:
\[ x̄ = \frac{\beta}{1 - \alpha} \quad \text{(5)} \]

7. **Equating the Two Expressions for \( x̄ \)**:
From equations (4) and (5), we have:
\[ \frac{b}{1 - a} = \frac{\beta}{1 - \alpha} \]

8. **Cross-Multiplying**:
Cross-multiplying gives:
\[ b(1 - \alpha) = \beta(1 - a) \]

9. **Finding the Ratio \( \frac{b}{\beta} \)**:
Rearranging the above equation leads to:
\[ \frac{b}{\beta} = \frac{1 - a}{1 - \alpha} \]

### Final Result:
Thus, the final result is:
\[ \frac{b}{\beta} = \frac{1 - a}{1 - \alpha} \]