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 plane \( XY \) (which is defined by \( z = 0 \)) in the ratio \( k : 1 \).

### Step-by-step Solution:

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

2. **Use the Section Formula**:
The coordinates of a point \( R \) that divides the line segment joining points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) in the ratio \( m:n \) are given by:
\[ R\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]
Here, we have \( m = k \) and \( n = 1 \).

3. **Calculate the Coordinates of Point R**:
Using the section formula, the coordinates of point \( R \) can be expressed as:
\[ R\left(\frac{k \cdot 2 + 1 \cdot 1}{k + 1}, \frac{k \cdot 3 + 1 \cdot (-1)}{k + 1}, \frac{k \cdot 4 + 1 \cdot 5}{k + 1}\right) \]
This simplifies to:
\[ R\left(\frac{2k + 1}{k + 1}, \frac{3k - 1}{k + 1}, \frac{4k + 5}{k + 1}\right) \]

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

5. **Solve for k**:
To solve the equation, we set the numerator equal to zero:
\[ 4k + 5 = 0 \]
Solving for \( k \):
\[ 4k = -5 \implies k = -\frac{5}{4} \]

### Final Answer:
The value of \( k \) is \( -\frac{5}{4} \).