If xz-plane divide the join of point `(2, 3, 4) and (1, -1, 5)` in the ratio `lambda:1`, then the integer `lambda` should be equal to

Answer

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

### Step-by-step Solution:

1. **Identify the coordinates of the points**:
- Let \( A = (2, 3, 4) \)
- Let \( B = (1, -1, 5) \)

2. **Understand the xz-plane**:
- The xz-plane is defined by \( y = 0 \). This means we want to find a point on the line segment \( AB \) where the y-coordinate is 0.

3. **Use the section formula**:
- The coordinates of the 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, \( (x_1, y_1, z_1) = (2, 3, 4) \) and \( (x_2, y_2, z_2) = (1, -1, 5) \).

4. **Substitute the coordinates into the formula**:
- The coordinates of point \( P \) become:
\[ P = \left( \frac{\lambda \cdot 1 + 2}{\lambda + 1}, \frac{\lambda \cdot (-1) + 3}{\lambda + 1}, \frac{\lambda \cdot 5 + 4}{\lambda + 1} \right) \]

5. **Set the y-coordinate to 0**:
- Since we want the y-coordinate of point \( P \) to be 0:
\[ \frac{\lambda \cdot (-1) + 3}{\lambda + 1} = 0 \]

6. **Solve for \( \lambda \)**:
- Setting the numerator to zero gives:
\[ \lambda \cdot (-1) + 3 = 0 \]
- Rearranging this gives:
\[ -\lambda + 3 = 0 \implies \lambda = 3 \]

7. **Conclusion**:
- The integer \( \lambda \) should be equal to \( 3 \).

### Final Answer:
\[ \lambda = 3 \]