<p>A plane which bisects the angle between the two given planes \(2x - y + 2z - 4 = 0\) and \(x + 2y + 2z - 2 = 0\), passes through the point:</p>
Step-by-Step Solution
Key Concept: The angle bisector planes between two planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 are given by (a₁x + b₁y + c₁z + d₁)/√(a₁² + b₁² + c₁²) = ±(a₂x + b₂y + c₂z + d₂)/√(a₂² + b₂² + c₂²). Any point satisfying either bisector plane equation lies on an angle bisector plane.
Step 1: Write the two given planes: P_1: 2x - y + 2z - 4 = 0 P_2: x + 2y + 2z - 2 = 0 Step 2: Find normal vectors and their magnitudes: For P_1: n_1 = (2, -1, 2), |n_1| = √(4 + 1 + 4) = 3 For P_2: n_2 = (1, 2, 2), |n_2| = √(1 + 4 + 4) = 3 Step 3: Write the angle bisector planes using the formula: (2x - y + 2z - 4)/3 = ±(x + 2y + 2z - 2)/3 Step 4: Simplify both cases: Case 1 (+): 2x - y + 2z - 4 = x + 2y + 2z - 2 ⟹ x - 3y - 2 = 0 Case 2 (−): 2x - y + 2z - 4 = −(x + 2y + 2z - 2) ⟹ 3x - 3y + 4z - 6 = 0 ⟹ x - y + (4/3)z - 2 = 0 Step 5: Test given options in both bisector plane equations. The point that satisfies one of these equations is the answer. ∴ Answer: B
Correct Answer: B