<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: To maximize |PR - QR|, point R must lie on the line through P and Q extended beyond one endpoint (the locus is a hyperboloid, but R is constrained to a plane). The maximum occurs when R lies on the line PQ extended, satisfying both the plane constraint and the hyperboloid condition.
Step 1: Verify P and Q lie on plane P_1: 2x + y - z = 3 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 direction vector of line PQ PQ = Q - P = (4, -2, 6), or simplified: (2, -1, 3) Step 3: Parametric equation of line PQ R(t) = P + t·PQ = (-1, 2, -3) + t(4, -2, 6) = (-1+4t, 2-2t, -3+6t) Step 4: Substitute into constraint plane x_0 - 2y_0 + 3z_0 + 1 = 0 (-1+4t) - 2(2-2t) + 3(-3+6t) + 1 = 0 -1 + 4t - 4 + 4t - 9 + 18t + 1 = 0 26t - 13 = 0 t = 1/2 Step 5: Find coordinates of R x_0 = -1 + 4(1/2) = 1 y_0 = 2 - 2(1/2) = 1 z_0 = -3 + 6(1/2) = 0 Step 6: Calculate x_0 + y_0 + z_0 x_0 + y_0 + z_0 = 1 + 1 + 0 = 2 ∴ Answer: 2
Correct Answer: B