Let P₁:r⃗ . ( 2î + ĵ - 3k̂ ) = 4 be a plane. Let P₂ be another plane which passes through the points (2, -3, 2), (2, -2, -3) and (1, -4, 2). If the direction ratios of the line of intersection of P₁ and P₂ be 16, α, β, then the value of α + β is equal to ________________.

Solution

Maths sir · 4 steps

The line lies in both planes, so it is perpendicular to both normals: cross the two normals.

1. Normal of P2. Take A(2,-3,2), B(2,-2,-3), C(1,-4,2). Two vectors in the plane are AB⃗ = (0, 1, -5) and AC⃗ = (-1, -1, 0). Their cross product is n⃗₂ = (-5, 5, 1).
2. Check it. n⃗₂ · AB⃗ = 0 + 5 - 5 = 0 and n⃗₂ · AC⃗ = 5 - 5 + 0 = 0, so n⃗₂ is a correct normal. The normal of P₁ is n⃗₁ = (2, 1, -3).

   n⃗₁ × n⃗₂ = |î, ĵ, k̂; 2, 1, -3; -5, 5, 1| = (1+15)î - (2-15)ĵ + (10+5)k̂

3. Direction ratios. This gives (16, 13, 15). The first entry is already 16, so α = 13 and β = 15.
4. So. α + β = 13 + 15 = 28.

The common slip: 2. Forgetting the minus sign on the ĵ term gives (16, -13, 15), so α + β comes out as 2 instead of 28.

The idea to keep. The line where two planes meet is perpendicular to both normals, so its direction is n⃗₁ × n⃗₂. Find each normal first, then take one cross product.

Answer from the NTA final answer key. Solution by PrepWiser.