Question

For Lines: L: [(x – 23)/(– 6)] = [(y – 19)/(– 4)] = [(z – 25)/3] and M: [(x – 12)/(– 9)] = [(y – 1)/4] = [(z – 5)/2] and P ∈ L, Q ∈ M, PQ ⊥ L and PQ ⊥ M, then PQ = ____

(A) √26

(B) (1/26)

(C) (1/√26)

(D) 26

Answer 1

The correct option (D) 26

Explanation:

Lines: L = [(x – 23)/(– 6)] = [(y – 19)/(– 4)] = [(z – 25)/3] = k₁ 

Let P ∈ L 

∴ P(– 6k₁ + 23, – 4k₁ + 19, 3k₁ + 25) ....(1) 

Line: M = [(x – 12)/(– 9)] = [(y – 1)/4] = [(z – 5) / 2] = k₂ 

Let Q ∈ M

∴ Q(– 9k₂ + 12, 4k₂ + 1, 2k₂ + 5) ....(2)

∴ vector PQ = (– 9k₂ + 6k₁ – 11, 4k₂ + 4k₁ – 18, 2k₂ – 3k₁ – 20)

also vector ℓ = (– 6, – 4, 3) and vector m = (– 9, 4, 2)

∴ vector PQ ∙ vector ℓ = 0 gives 44k₂ – 61k₁ + 78 = 0 ......(3) 

∴ vector PQ ∙ vector m = 0 gives 101k₂ – 44k₁ – 13 = 0 .......(4)

solving (3) & (4)

k₁ = 2 and k₂ = 1

∴ from (1), P(11, 11, 31)

from (2), Q(3, 5, 7)

∴ vector PQ = (– 8, – 6, – 24)

∴ vector |PQ| = √(64 + 36 + 576) = √(676) = 26 unit.