The ratio in which the xy - plane divides the join of (1, 2, 3) and (4, 2, 1) is

Options

A. `3:1` internally
B. `3:1` externally
C. `1:2` internally
D. `2:1` externally

Answer

To find the ratio in which the xy-plane divides the line segment joining the points \( A(1, 2, 3) \) and \( B(4, 2, 1) \), we can use the section formula. The xy-plane is defined by the equation \( z = 0 \).

### Step 1: Identify the coordinates of the points
The coordinates of the points are:
- \( A(1, 2, 3) \)
- \( B(4, 2, 1) \)

### Step 2: Set up the section formula
Let the ratio in which the xy-plane divides the line segment \( AB \) be \( m:n \). According to the section formula, the coordinates of the point \( P \) dividing the line segment \( AB \) in the ratio \( m:n \) are given by:

\[ P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

Where:
- \( (x_1, y_1, z_1) = (1, 2, 3) \)
- \( (x_2, y_2, z_2) = (4, 2, 1) \)

### Step 3: Set the z-coordinate to 0
Since we want the point \( P \) to lie on the xy-plane, we set the z-coordinate to 0:

\[ \frac{mz_2 + nz_1}{m+n} = 0 \]

Substituting the z-coordinates of points \( A \) and \( B \):

\[ \frac{m \cdot 1 + n \cdot 3}{m+n} = 0 \]

### Step 4: Solve for the ratio \( m:n \)
This simplifies to:

\[ m \cdot 1 + n \cdot 3 = 0 \]

Rearranging gives:

\[ m + 3n = 0 \]

From this, we can express \( m \) in terms of \( n \):

\[ m = -3n \]

### Step 5: Find the ratio \( \frac{m}{n} \)
Now, we can find the ratio \( \frac{m}{n} \):

\[ \frac{m}{n} = \frac{-3n}{n} = -3 \]

### Step 6: Express the ratio in positive terms
The negative sign indicates that the division is external. Thus, we express the ratio as:

\[ m:n = 3:1 \]

### Conclusion
The xy-plane divides the line segment joining the points \( (1, 2, 3) \) and \( (4, 2, 1) \) in the ratio \( 3:1 \) externally.