The plane-XY divides the join of (1, -1, 5) and
(2, 3, 4) in the ratio `k : 1` then k is

Options

A. `5/4`
B. `-5/4`
C. `3/2`
D. `4/3`

Answer

To solve the problem, we need to find the value of \( k \) such that the line segment joining the points \( P(1, -1, 5) \) and \( Q(2, 3, 4) \) is divided by the XY-plane in the ratio \( k:1 \).

### Step-by-Step Solution:

1. **Identify the Coordinates of Points**:
- Let \( P(1, -1, 5) \) be the first point.
- Let \( Q(2, 3, 4) \) be the second point.

2. **Use the Section Formula**:
- The coordinates of the point \( R \) that divides the line segment joining points \( P \) and \( Q \) in the ratio \( k:1 \) can be calculated using the section formula:
\[ R\left(\frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1}\right) \]
- Here, \( (x_1, y_1, z_1) = (1, -1, 5) \) and \( (x_2, y_2, z_2) = (2, 3, 4) \).

3. **Calculate the Coordinates of Point R**:
- The x-coordinate of \( R \):
\[ x_R = \frac{k \cdot 2 + 1 \cdot 1}{k + 1} = \frac{2k + 1}{k + 1} \]
- The y-coordinate of \( R \):
\[ y_R = \frac{k \cdot 3 + 1 \cdot (-1)}{k + 1} = \frac{3k - 1}{k + 1} \]
- The z-coordinate of \( R \):
\[ z_R = \frac{k \cdot 4 + 1 \cdot 5}{k + 1} = \frac{4k + 5}{k + 1} \]

4. **Set the z-coordinate to 0**:
- Since the point \( R \) lies on the XY-plane, we set \( z_R = 0 \):
\[ \frac{4k + 5}{k + 1} = 0 \]

5. **Solve for k**:
- The numerator must be zero for the fraction to equal zero:
\[ 4k + 5 = 0 \]
- Solving for \( k \):
\[ 4k = -5 \implies k = -\frac{5}{4} \]

6. **Interpret the Result**:
- The negative value of \( k \) indicates that the division of the segment is external, meaning point \( R \) lies outside the segment \( PQ \).

### Final Answer:
\[ k = -\frac{5}{4} \]