Question

If the plane \[ x - y + z + 4 = 0 \] divides the line joining the points \[ P(2,3,-1) \quad \text{and} \quad Q(1,4,-2) \] in the ratio \( l:m \), then \( l + m \) is:

• A. \( -1 \)
• B. \( 3 \)
• C. \( -1 \)
• D. \( 4 \)

Hint:

For a plane dividing a line segment, apply the section formula in 3D and substitute into the given plane equation.

Answer 1

The Correct Option is B

Step 1: Section formula in 3D Using the section formula, the point dividing \( P \) and \( Q \) in the ratio \( l:m \) is: \[ (x, y, z) = \left(\frac{l x_2 + m x_1}{l+m}, \frac{l y_2 + m y_1}{l+m}, \frac{l z_2 + m z_1}{l+m} \right). \] Solving for the given plane equation, we find: \[ l + m = 3. \]