The line of regression Y and X referred to `barX,barY` as origin is

Options

A. `Y-barY=r(sigma_(Y))/(sigma_(X))(X-barX)`
B. `Y=r(sigma_(Y))/(sigma_(X))(X-barX)`
C. `Y=r(sigma_(Y))/(sigma_(X))X`
D. none of these

Answer

To solve the question regarding the line of regression \( Y \) and \( X \) with respect to the point \( (\bar{X}, \bar{Y}) \) as the origin, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Origin**:
Since \( (\bar{X}, \bar{Y}) \) is considered the origin, we have:
\[ \bar{X} = 0 \quad \text{and} \quad \bar{Y} = 0 \]

2. **Regression Equation**:
The line of regression of \( Y \) on \( X \) can be expressed in the form:
\[ Y - \bar{Y} = B_{YX}(X - \bar{X}) \]
where \( B_{YX} \) is the regression coefficient of \( Y \) on \( X \).

3. **Substituting the Origin Values**:
Since \( \bar{X} = 0 \) and \( \bar{Y} = 0 \), we can substitute these values into the regression equation:
\[ Y - 0 = B_{YX}(X - 0) \]
This simplifies to:
\[ Y = B_{YX} \cdot X \]

4. **Finding the Regression Coefficient**:
The regression coefficient \( B_{YX} \) is defined as:
\[ B_{YX} = r \cdot \frac{\sigma_Y}{\sigma_X} \]
where \( r \) is the correlation coefficient, \( \sigma_Y \) is the standard deviation of \( Y \), and \( \sigma_X \) is the standard deviation of \( X \).

5. **Substituting the Regression Coefficient**:
Now, substituting \( B_{YX} \) into the equation we derived:
\[ Y = r \cdot \frac{\sigma_Y}{\sigma_X} \cdot X \]

6. **Final Equation**:
Thus, the final equation of the line of regression \( Y \) on \( X \) is:
\[ Y = r \cdot \frac{\sigma_Y}{\sigma_X} \cdot X \]

### Conclusion:
The correct option that matches our derived equation is the third option:
\[ Y = r \cdot \frac{\sigma_Y}{\sigma_X} \cdot X \]