Let a⃗=î+2 ĵ+k̂ and b⃗=2 î+7 ĵ+3 k̂. Let L₁ : r⃗=(-î+2 ĵ+k̂)+λ a⃗, λ ∈ R and L₂: r⃗=(ĵ+k̂)+μ b⃗, μ ∈ R be two lines. If the line L₃ passes through the point of intersection of L₁ and L_y and is parallel to a⃗+b⃗, then L₃ passes through the point :

Solution

Maths sir · 3 steps

Find the meeting point of L₁ and L₂ first, then walk along a⃗+b⃗ until you hit an option.

1. Equate the lines. A general point on L₁ is (-1+λ, 2+2λ, 1+λ) and on L₂ is (2μ, 1+7μ, 1+3μ). The z-coordinates give λ = 3μ. The x-coordinates then give -1+3μ = 2μ, so μ = 1 and λ = 3.
2. Intersection point. Check the y-coordinates: 2+2(3) = 8 and 1+7(1) = 8, so the lines really meet. The point is (2, 8, 4).

   L₃: (x, y, z) = (2, 8, 4) + t (3, 9, 4), a⃗+b⃗ = 3î+9ĵ+4k̂

   Option	t from x	Point on L₃ at that t	Matches?
   (-1, -1, 1)	-1	(-1, -1, 0)	no, z differs
   (2, 8, 5)	0	(2, 8, 4)	no, z differs
   (8, 26, 12)	2	(8, 26, 12)	yes
   (5, 17, 4)	1	(5, 17, 8)	no, z differs

3. So. At t = 2, L₃ passes through (8, 26, 12), which is option (c).

The idea to keep. To test whether a point lies on a line, find the parameter from one coordinate and check that the same value works for all three.

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