The xy-plane divides the line joining the points (-1,3,4) aned (2,-5,6)

Options

A. internally in the ratio 2:3
B. externally in the ratio 2:3
C. internally in the ratio 3:2
D. externally in the ratio 3:2

Explanation

To solve the problem of how the xy-plane divides the line segment joining the points (-1, 3, 4) and (2, -5, 6), we will use the section formula in three-dimensional geometry.

### Step-by-Step Solution:

1. **Identify the Points**:
Let the points be:
- Point A: \( A(-1, 3, 4) \)
- Point B: \( B(2, -5, 6) \)

2. **Set Up the Section Formula**:
The section formula in 3D states that if a point \( P(x, y, z) \) divides the line segment joining points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]

3. **Determine the z-coordinate**:
Since the xy-plane is defined by \( z = 0 \), we set the z-coordinate of point \( P \) to 0:
\[ \frac{m \cdot 6 + n \cdot 4}{m+n} = 0 \]

4. **Solve for the Ratio**:
Rearranging the equation gives:
\[ m \cdot 6 + n \cdot 4 = 0 \]
This can be rewritten as:
\[ 6m + 4n = 0 \]
From this, we can express \( m \) in terms of \( n \):
\[ 6m = -4n \quad \Rightarrow \quad m = -\frac{2}{3}n \]

5. **Express the Ratio**:
The ratio \( m:n \) can be expressed as:
\[ m:n = -\frac{2}{3}n : n = -2:3 \]
Since we are interested in the ratio of division, we take the absolute values:
\[ \text{Ratio} = 2:3 \]

6. **Determine the Type of Division**:
Since \( m \) is negative, it indicates that the division is external. Therefore, the xy-plane divides the line segment externally in the ratio \( 2:3 \).

### Final Answer:
The xy-plane divides the line joining the points (-1, 3, 4) and (2, -5, 6) externally in the ratio \( 2:3 \).