Two points on a line are such that their Y co-ordinates is `(3 + sqrt(2))` times the X co-ordinate. Then the line

Options

A. has negative slope
B. passes through the origin
C. is parallel to the X-axis
D. is parallel to the Y-axis

Answer

To solve the problem, we need to find the equation of the line given that the y-coordinates of two points on the line are `(3 + sqrt(2))` times their respective x-coordinates.

### Step-by-Step Solution:

1. **Define the Points**:
Let the two points on the line be \( A(x_1, y_1) \) and \( B(x_2, y_2) \).

2. **Express y-coordinates**:
According to the problem, the y-coordinates can be expressed as:
\[ y_1 = (3 + \sqrt{2}) x_1 \]
\[ y_2 = (3 + \sqrt{2}) x_2 \]

3. **Equation of the Line**:
The general equation of a line can be written as:
\[ y = mx + c \]
where \( m \) is the slope and \( c \) is the y-intercept.

4. **Substituting Points into the Line Equation**:
Since both points lie on the line, they must satisfy the line equation. Thus, we can write:
\[ (3 + \sqrt{2}) x_1 = mx_1 + c \quad \text{(Equation 1)} \]
\[ (3 + \sqrt{2}) x_2 = mx_2 + c \quad \text{(Equation 2)} \]

5. **Subtracting the Equations**:
To eliminate \( c \), we subtract Equation 2 from Equation 1:
\[ (3 + \sqrt{2}) x_1 - (3 + \sqrt{2}) x_2 = mx_1 - mx_2 \]
This simplifies to:
\[ (3 + \sqrt{2})(x_1 - x_2) = m(x_1 - x_2) \]

6. **Finding the Slope (m)**:
If \( x_1 \neq x_2 \), we can divide both sides by \( (x_1 - x_2) \):
\[ m = 3 + \sqrt{2} \]

7. **Finding the y-intercept (c)**:
Now we substitute \( m \) back into one of the original equations to find \( c \). Let's use Equation 1:
\[ (3 + \sqrt{2}) x_1 = (3 + \sqrt{2}) x_1 + c \]
Rearranging gives:
\[ c = 0 \]

8. **Final Equation of the Line**:
Now we can write the equation of the line:
\[ y = (3 + \sqrt{2}) x \]

### Conclusion:
Since the y-intercept \( c = 0 \), the line passes through the origin.

### Answer:
The line passes through the origin (Option B).