Let S be the set of all λ , μ for which the vectors λ i ^ - j ^ + k ^ , j ^ + 2 j ^ + μ k ^ and 3 i ^ - 4 j ^ + 5 k ^ , where λ - μ = 5 , are coplanar, then ∑ λ , μ ∈ S 80 λ 2 + μ 2 is equal to

Step-by-step solution

The given equations are coplanar. ⇒   λ - 1 1 1 2 μ 3 - 4 5 = 0 ⇒ λ 10 + 4 μ + 1 5 - 3 μ + 1 - 10 = 0 ⇒ 4 λ μ + 10 λ - 3 μ = 5   . . . . . . . i And λ - μ = 5 So, by 1 ⇒ 4 5 + μ μ + 10 5 + μ - 3 μ = 5 ⇒ 4 μ 2 + 27 μ + 45 = 0 ⇒ 4 μ 2 + 15 μ + 12 μ + 45 = 0 ∴   μ = - 3 ,   - 15 4 ⇒ λ = 2 ,   5 4 ∴   ∑ λ μ &#871