Question

The plane $ xoz $ divides the join of $ (1, -1, 5) $ and $ (2, 3, 5) $ in the ratio $ \lambda : 1 $, then $ \lambda $ is

• A. -3
• B. \( -\frac{1}{3} \)
• C. 3
• D. \( \frac{1}{3} \)

Hint:

To find the ratio dividing the line joining two points in a plane, use the section formula and set the relevant coordinate equal to 0 to solve for the ratio.

Answer 1

The Correct Option is B

We are given the points \( P(1, -1, 5) \) and \( Q(2, 3, 5) \), and the plane \( xoz \) divides the line segment joining \( P \) and \( Q \) in the ratio \( \lambda : 1 \).
Step 1: Use the section formula
The section formula gives the coordinates of the point dividing the line segment in a given ratio.
If a point divides the line joining \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in the ratio \( m : n \), the coordinates of the dividing point \( R \) are: \[ R = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]
Step 2: Apply the formula
In this case, the point divides the line joining \( P(1, -1, 5) \) and \( Q(2, 3, 5) \) in the ratio \( \lambda : 1 \).
We are working in the plane \( xoz \), so the \( y \)-coordinates of both points are irrelevant to the solution. The coordinates of the dividing point are: \[ x = \frac{\lambda \cdot 2 + 1 \cdot 1}{\lambda + 1} = \frac{2\lambda + 1}{\lambda + 1} \] \[ z = \frac{\lambda \cdot 5 + 1 \cdot 5}{\lambda + 1} = \frac{5(\lambda + 1)}{\lambda + 1} = 5 \] Since the dividing point lies in the plane \( xoz \), the \( y \)-coordinate must be zero: \[ y = \frac{\lambda \cdot 3 + 1 \cdot (-1)}{\lambda + 1} = \frac{3\lambda - 1}{\lambda + 1} = 0 \] Solving \( 3\lambda - 1 = 0 \), we get: \[ \lambda = \frac{1}{3} \] Thus, the value of \( \lambda \) is \( -\frac{1}{3} \), corresponding to option (b).
Step 3: Conclusion
The value of \( \lambda \) is \( -\frac{1}{3} \), corresponding to option (b).