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In three-dimensional space, consider the planes P1: y = 0 and P2: x + z = 1. Let P3 represent a plane distinct from both P1 and P2, passing through the line formed by the intersection of P1 and P2. If the point (0, 1, 0) lies at a distance of 1 unit from P3, and the point (α, β, γ) lies at a distance of 2 units from P3, which of the following equations is/are valid?

1. 2α + β + 2γ + 2 = 0
2. 2α - β + 2γ + 4 = 0
3. 2α + β - 2γ - 10 = 0
4. 2α - β + 2γ - 8 = 0

Correct answer: 2α - β + 2γ + 4 = 0

Solution

The plane P₃ passes through the intersection of P₁ and P₂ and satisfies the given distance conditions. Substituting the point (α, β, γ) into the plane equations confirms that only 2α - β + 2γ + 4 = 0 is valid.

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