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Solution


Q.

Let O be the orgin, and M and N be the points on the lines x–54=y–41=z–53 and x+812=y+25=z+119 respectively such that MN is the shortest distance between the given lines. Then OM→·ON→ is equal to __________.          [2024]

Ans.

(9)

Let, L1 : x–54=y–41=z–53=λ∈R

L2 : x+812=y+25=z+119=μ∈R

M is a point on L1.

So, M=(4λ+5, λ+4, 3λ+5)

and N is a point on L2.

So, N=(12μ–8, 5μ–2, 9μ–11)

The direction ratios of MN is

(12μ–4λ–13, 5μ–λ–6, 9μ–3λ–16)

∵  MN⊥L1 and MN⊥L2

∴  4(12μ–4λ–13)+5μ–λ–6+3(9μ–3λ–16)=0

⇒48μ–16λ–52+5μ–λ-6+27μ–9λ–48=0

⇒80μ–26λ–106=0 ⇒ 40μ–13λ–53=0          ... (i)

Also,

12(12μ–4λ–13)+5(5μ–λ–6)+9(9μ–3λ–16)=0

144μ–48λ–156+25μ–5λ–30+81μ–27λ–144=0

⇒ 250μ–80λ–330=0⇒ 25μ–8λ–33=0          ... (ii)

On solving equation (i) and (ii), we get μ=1 and λ=–1

M = (1, 3, 2) and N = (4, 3, –2)

∴  OM→ =i^+3j^+2k^ and ON→ =4i^+3j^–2k^

    OM→·ON→ = 4 + 9 – 4 = 9.