Let S be the set of all (λ, μ) for which the vectors λ î-ĵ+k̂, î+2 ĵ+μ k̂ and 3 î-4 ĵ+5 k̂, where λ-μ=5, are coplanar, then Σ_((λ, μ) ∈ S) 80(λ²+μ²) is equal to :

Solution

Maths sir · 4 steps

Coplanar means the determinant is zero. Put λ = μ + 5 in it and solve one quadratic.

1. Coplanarity condition. Three vectors are coplanar exactly when their scalar triple product is zero. That is the determinant of their components.

   |λ, -1, 1; 1, 2, μ; 3, -4, 5| = λ(10+4μ) + (5-3μ) - 10 = 4λμ + 10λ - 3μ - 5 = 0

2. Use the constraint. Put λ = μ + 5: 4μ(μ+5) + 10(μ+5) - 3μ - 5 = 0, which gives 4μ² + 27μ + 45 = 0. The roots are μ = (-27 ± 3)/8, that is, μ = -3 or μ = -15/4.
3. The two pairs. μ = -3 gives λ = 2, so λ² + μ² = 4 + 9 = 13. μ = -15/4 gives λ = 5/4, so λ² + μ² = 25/16 + 225/16 = 250/16.
4. So. 80(13 + 250/16) = 1040 + 1250 = 2290, which is option (d).

The idea to keep. For coplanarity, set the 3 by 3 determinant of the components to zero. When a linear constraint links the unknowns, substitute it first, and you are left with one equation in one variable.

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