If the points with position vectors α î+10 ĵ+13 k̂, 6 î+11 ĵ+11 k̂, 9/2 î+β ĵ-8 k̂ are collinear, then (19 α-6 β)² is equal to :

Solution

Maths sir · 4 steps

Collinear means BC is a multiple of AB, and the k̂ parts hand you that multiple.

1. Two direction vectors. Call the points A, B, C. Then AB⃗ = (6-α)î + ĵ - 2k̂ and BC⃗ = -3/2î + (β - 11)ĵ - 19k̂.
2. Find the multiple. For collinear points, BC⃗ = λ AB⃗. The k̂ components have no unknowns: -19 = -2λ, so λ = 19/2.

   -3/2 = 19/2(6-α), β - 11 = 19/2

3. Solve each. From the first, 6 - α = -3/19, so α = 117/19 and 19α = 117. From the second, β = 41/2 and 6β = 123.
4. So. 19α - 6β = 117 - 123 = -6, so (19α - 6β)² = 36. The answer is 36, option (c).

The idea to keep. For collinearity, write two difference vectors and set one equal to λ times the other. Start with the component that has no unknowns, because it gives λ straight away.

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