Ratio in which the xy-plane divides the joint 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

Explanation

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 first need to determine the coordinates of the points where the line segment intersects the xy-plane. The xy-plane is defined by the equation \( z = 0 \).

### Step 1: Find the parametric equations of the line segment
The parametric equations of the line segment joining points \( A \) and \( B \) can be expressed as:
\[ x = x_1 + t(x_2 - x_1) = 1 + t(4 - 1) = 1 + 3t \]
\[ y = y_1 + t(y_2 - y_1) = 2 + t(2 - 2) = 2 \]
\[ z = z_1 + t(z_2 - z_1) = 3 + t(1 - 3) = 3 - 2t \]
where \( t \) varies from \( 0 \) to \( 1 \).

### Step 2: Set \( z = 0 \) to find the intersection with the xy-plane
To find the point where the line intersects the xy-plane, we set \( z = 0 \):
\[ 3 - 2t = 0 \]
Solving for \( t \):
\[ 2t = 3 \implies t = \frac{3}{2} \]

### Step 3: Find the coordinates of the intersection point
Now we substitute \( t = \frac{3}{2} \) back into the equations for \( x \) and \( y \):
\[ x = 1 + 3\left(\frac{3}{2}\right) = 1 + \frac{9}{2} = \frac{11}{2} \]
\[ y = 2 \]
Thus, the intersection point is \( \left( \frac{11}{2}, 2, 0 \right) \).

### Step 4: Determine the ratio in which the xy-plane divides the segment
The ratio in which the segment is divided can be found using the formula:
\[ \text{Ratio} = \frac{t}{1-t} \]
Substituting \( t = \frac{3}{2} \):
\[ \text{Ratio} = \frac{\frac{3}{2}}{1 - \frac{3}{2}} = \frac{\frac{3}{2}}{-\frac{1}{2}} = -3 \]
This indicates that the xy-plane divides the segment in the ratio \( 3:1 \) (the negative sign indicates that the division is in the opposite direction).

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