XY-plane divides the line joining the points (2, 4, 5)
and (-4, 3, -2) in the ratio

Options

A. `3 : 5`
B. `5 : 2`
C. `1 : 3`
D. `3 : 4`

Answer

To find the ratio in which the XY-plane divides the line segment joining the points \( A(2, 4, 5) \) and \( B(-4, 3, -2) \), we can use the section formula. The XY-plane is defined by the equation \( z = 0 \).

### Step-by-step Solution:

1. **Identify the Points**:
Let \( A(2, 4, 5) \) and \( B(-4, 3, -2) \).

2. **Assume the Point of Division**:
Let the point \( R \) divide the line segment \( AB \) in the ratio \( \lambda : 1 \).

3. **Use the Section Formula**:
The coordinates of point \( R \) can be calculated using the section formula:
\[ R = \left( \frac{\lambda x_2 + x_1}{\lambda + 1}, \frac{\lambda y_2 + y_1}{\lambda + 1}, \frac{\lambda z_2 + z_1}{\lambda + 1} \right) \]
where \( (x_1, y_1, z_1) = (2, 4, 5) \) and \( (x_2, y_2, z_2) = (-4, 3, -2) \).

4. **Substituting the Coordinates**:
Substituting the coordinates of points \( A \) and \( B \) into the formula gives:
\[ R = \left( \frac{\lambda(-4) + 2}{\lambda + 1}, \frac{\lambda(3) + 4}{\lambda + 1}, \frac{\lambda(-2) + 5}{\lambda + 1} \right) \]

5. **Setting the z-coordinate to 0**:
Since \( R \) lies on the XY-plane, we set the z-coordinate to 0:
\[ \frac{\lambda(-2) + 5}{\lambda + 1} = 0 \]

6. **Solving for \(\lambda\)**:
This implies:
\[ \lambda(-2) + 5 = 0 \]
\[ -2\lambda + 5 = 0 \]
\[ 2\lambda = 5 \]
\[ \lambda = \frac{5}{2} \]

7. **Finding the Ratio**:
The ratio in which the XY-plane divides the line segment \( AB \) is \( \lambda : 1 = \frac{5}{2} : 1 = 5 : 2 \).

### Conclusion:
Thus, the XY-plane divides the line joining the points \( (2, 4, 5) \) and \( (-4, 3, -2) \) in the ratio \( 5 : 2 \).