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Solution


Q.

Let a line passing through the point (–1, 2, 3) intersect the lines L1:x–13=y–22=z+1–2 at M(α,β,γ) and L2:x+2–3=y–2–2=z–14 at N(a, b, c). Then the value of (α+β+γ)2(a+b+c)2 equals __________.           [2024]

Ans.

(196)

We have, L1 : x–13=y–22=z+1–2=λ∈R

∴  M≡(3λ+1,2λ+2,–2λ–1)

L2 : x+2–3=y–2–2=z–14=μ∈R

∴  N≡(–3μ—2,–2μ+2,4μ+1)

α+β+γ=3λ+2 ⇒ a+b+c=–μ+1

Direction ratio's of line AM = <3λ+2,2λ,–2λ–4>

Direction ratio's of line AN = <–3μ—1,–2μ,4μ–2>

⇒ 3λ+2–3μ–1=2λ–2μ=–2λ–44μ–2

⇒ 3λ+2–3μ–1=λ–μ and λ–μ=–2λ–44μ–2

⇒ –3λμ–2μ=–3μλ–λ and 4μλ–2λ=2μλ+4μ

⇒ 2μ=λ and 2μλ=2λ+4μ

⇒ λ2=2λ+2λ ⇒ λ2=4λ

⇒ λ(λ–4)=0 ⇒ λ=0, λ=4

∴  (α+β+γ)2(a+b+c)2=(3×4+2)2(–2+1)2=(14)21=196.