Let L : y-2 y-4 and L : be two lines. Then which of the following points lies on x-1 z-3 x-2 z-5 1 = = 2 = = 2 3 4 3 4 5 the line of the shortest distance between L and L ? 1 2
Step-by-Step Solution
Key Concept: Apply the core result for lines and planes in three dimensions and simplify using the given constraints.
L1 : x - 1 = y - 2 = z - 3 2 3 4 (1) x - 2 y - 4 z - 5 L2 : = = 3 4 5 P (2\lambda + 1, 3\lambda + 2, 4\lambda + 3) Q(3\mu + 2, 4\mu + 4, 5\mu + 5) Dr's of P Q < 2\lambda - 3\mu - 1, 3\lambda - 4\mu - 2, 4\lambda - 5\mu - 2 > ∣^ ^ ^∣ i j k ∣ ∣ ^ ^ ^ PQ = ∣ 2 3 4 ∣ = - i + 2j - k ∣ ∣ ∣3 4 5∣ 2\lambda - 3\mu - 1 3\lambda - 4\mu - 2 4\lambda - 5\mu - 2 \Rightarrow = = -1 2 -1 1 -1 \Rightarrow \lambda = \mu = 3 6 5 13 3 10 25 \Rightarrow P ( , 3, ) Q( , , ) 3 3 2 3 6 Dr's P Q⟨1, -2, 1⟩ \therefore Line 5 13 y- y- 3 y-3 3 = = 1 -2 1
Correct Answer: 1