<p>If \(P(-1, 2, -3)\) and \(Q(3, 0, 3)\) are two points on the plane \(P_1: 2x + y - z = 3\) and \(R(x_0, y_0, z_0)\) be a point such that \(x_0 - 2y_0 + 3z_0 + 1 = 0\) and \(|PR - QR|\) is maximum, then \((x_0 + y_0 + z_0)\) is equal to:</p>
Step-by-Step Solution
Key Concept: The maximum value of |PR - QR| occurs when R lies on the line through P and Q extended beyond one of them. Since R is constrained to a plane, find where the line PQ intersects the constraint plane—this gives the extremal point.
Step 1: Verify P and Q lie on plane P_1 For P(-1, 2, -3): 2(-1) + 2 - (-3) = -2 + 2 + 3 = 3 ✓ For Q(3, 0, 3): 2(3) + 0 - 3 = 6 - 3 = 3 ✓ Step 2: Find the line PQ Direction vector: Q - P = (4, -2, 6) or simplified (2, -1, 3) Parametric form: (x, y, z) = (-1, 2, -3) + t(2, -1, 3) = (-1+2t, 2-t, -3+3t) Step 3: Recognize the extremal condition The maximum of |PR - QR| equals |PQ| when R lies on the line PQ extended (collinear). This occurs when R is at the intersection of line PQ with the constraint plane x_0 - 2y_0 + 3z_0 + 1 = 0. Step 4: Find intersection point Substitute parametric equations into constraint: (-1+2t) - 2(2-t) + 3(-3+3t) + 1 = 0 -1 + 2t - 4 + 2t - 9 + 9t + 1 = 0 13t - 13 = 0 t = 1 Step 5: Calculate R R = (-1+2(1), 2-1, -3+3(1)) = (1, 1, 0) Step 6: Find the sum x_0 + y_0 + z_0 = 1 + 1 + 0 = 2 ∴ Answer: B (2)
Correct Answer: B