Question

The equation of the plane through the intersection of the planes \( 3x - y + 2z - 4 = 0 \) and \( x + y + z - 2 = 0 \) and the point \( (2,2,1) \) is

• A. \( 7x + 5y + 4z + 8 = 0 \)
• B. \( 7x + 5y + 4z - 8 = 0 \)
• C. \( 7x - 5y + 4z - 8 = 0 \)
• D. None of these

Hint:

Use \( P_1 + \lambda P_2 = 0 \) to find plane through intersection, then determine \( \lambda \) using given point.

Answer 1

The Correct Option is C

Concept: Equation of plane through intersection of two planes: \[ P_1 + \lambda P_2 = 0 \]

Step 1: Form general equation. \[ (3x - y + 2z - 4) + \lambda(x + y + z - 2) = 0 \] \[ (3+\lambda)x + (-1+\lambda)y + (2+\lambda)z - (4+2\lambda) = 0 \]

Step 2: Substitute point \( (2,2,1) \): \[ (3+\lambda)2 + (-1+\lambda)2 + (2+\lambda)1 - (4+2\lambda) = 0 \] \[ 6+2\lambda -2+2\lambda +2+\lambda -4-2\lambda = 0 \] \[ 2 + 3\lambda = 0 \Rightarrow \lambda = -\frac{2}{3} \]

Step 3: Substitute \( \lambda \): \[ \frac{7}{3}x - \frac{5}{3}y + \frac{4}{3}z - \frac{8}{3} = 0 \] Multiply by 3: \[ 7x - 5y + 4z - 8 = 0 \] Final Answer: \[ 7x - 5y + 4z - 8 = 0 \]