3D Geometry
Family of Planes
Grade 12
Question:
<p>The equation of a plane containing the line of intersection of the planes \(2x - y - 4 = 0\) and \(y + 2z - 4 = 0\) and passing through the point \((1, 1, 0)\) is:</p>
<p>\(x - 3y - 2z = -2\)</p>
<p>\(2x - z = 2\)</p>
<p>\(x - y - z = 0\)</p>
<p>\(x + 3y + z = 4\)</p>
Step-by-Step Solution
Key Concept: A plane containing the line of intersection of two planes can be expressed as a linear combination of those planes: λ(P₁) + μ(P₂) = 0. Substitute the given point to find the ratio λ:μ.
Step 1: Express the family of planes containing the line of intersection as: λ(2x - y - 4) + μ(y + 2z - 4) = 0 Expanding: 2λx + (μ - λ)y + 2μz - 4λ - 4μ = 0 Step 2: Since the plane passes through point (1, 1, 0), substitute x = 1, y = 1, z = 0: λ(2(1) - 1 - 4) + μ(1 + 2(0) - 4) = 0 λ(-3) + μ(-3) = 0 -3λ - 3μ = 0 ⟹ λ = -μ Step 3: Choose λ = 1, μ = -1. Substitute back: 1(2x - y - 4) - 1(y + 2z - 4) = 0 2x - y - 4 - y - 2z + 4 = 0 2x - 2y - 2z = 0 Step 4: Simplify by dividing by 2: x - y - z = 0 ∴ Answer: C
Correct Answer: C