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

Options

A. `-3`
B. `-(1)/(3)`
C. `3`
D. `(1)/(3)`

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the plane \( XOZ \) divides the line segment joining the points \( A(1, -1, 5) \) and \( B(2, 3, 4) \) in the ratio \( \lambda:1 \).

### Step-by-Step Solution:

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

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

3. **Substituting the Values**:
- Substitute the coordinates into the formula:
\[ P\left( \frac{\lambda \cdot 2 + 1}{\lambda + 1}, \frac{\lambda \cdot 3 - 1}{\lambda + 1}, \frac{\lambda \cdot 4 + 5}{\lambda + 1} \right) \]
- This gives us:
\[ P\left( \frac{2\lambda + 1}{\lambda + 1}, \frac{3\lambda - 1}{\lambda + 1}, \frac{4\lambda + 5}{\lambda + 1} \right) \]

4. **Condition for the Plane \( XOZ \)**:
- The plane \( XOZ \) implies that the \( y \)-coordinate of point \( P \) must be zero:
\[ \frac{3\lambda - 1}{\lambda + 1} = 0 \]

5. **Solving for \( \lambda \)**:
- Set the numerator equal to zero:
\[ 3\lambda - 1 = 0 \]
- Solving for \( \lambda \):
\[ 3\lambda = 1 \implies \lambda = \frac{1}{3} \]

6. **Conclusion**:
- The value of \( \lambda \) is \( \frac{1}{3} \).