Question 35 (A) - CBSE Class 12 Sample Paper for 2025 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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Question 35 (A) Find the shortest distance between the lines 𝑙_1 and 𝑙_2 whose vector equations are 𝑟 ⃗=(−ı ˆ−ȷ ˆ−𝑘 ˆ)+𝜆(7ı ˆ−6ȷ ˆ+𝑘 ˆ)" and " 𝑟 ⃗=(3ı ˆ+5ȷ ˆ+7𝑘 ˆ)+𝜇(ı ˆ−2ȷ ˆ+𝑘 ˆ) where 𝜆 and 𝜇 are parameters.Shortest distance between the lines with vector equations 𝑟 ⃗ = (𝑎1) ⃗ + 𝜆 (𝑏1) ⃗and 𝑟 ⃗ = (𝑎2) ⃗ + 𝜇(𝑏2) ⃗ is |(((𝒃𝟏) ⃗ × (𝒃𝟐) ⃗ ).((𝒂𝟐) ⃗ − (𝒂𝟏) ⃗ ))/|(𝒃𝟏) ⃗ × (𝒃𝟐) ⃗ | | 𝒓 ⃗ = (–𝒊 ̂ – 𝒋 ̂ – 𝒌 ̂) + 𝜆(7𝒊 ̂ − 6𝒋 ̂ + 𝒌 ̂) Comparing with 𝑟 ⃗ = (𝑎1) ⃗ + 𝜆 (𝑏1) ⃗, (𝑎1) ⃗ = 𝑖 ̂ – 𝑗 ̂ – 𝑘 ̂ & (𝑏1) ⃗ = 7𝑖 ̂ – 6𝑗 ̂ + 𝑘 ̂ 𝒓 ⃗ = (3𝒊 ̂ + 5𝒋 ̂ + 7𝒌 ̂) + 𝝁 (𝒊 ̂ – 2𝒋 ̂ + 𝒌 ̂) Comparing with 𝑟 ⃗ = (𝑎2) ⃗ + 𝜇(𝑏2) ⃗ , (𝑎2) ⃗ = 3𝑖 ̂ + 5𝑗 ̂ + 7𝑘 ̂ & (𝑏2) ⃗ = 𝑖 ̂ – 2𝑗 ̂ + 𝑘 ̂ Now, (𝒂𝟐) ⃗ − (𝒂𝟏) ⃗ = (3𝑖 ̂ + 5𝑗 ̂ + 7𝑘 ̂) − (–𝑖 ̂ – 𝑗 ̂ – 𝑘 ̂) = (3 + 1) 𝑖 ̂ + (5 + 1)𝑗 ̂ + (7 + 1) 𝑘 ̂ = 4𝒊 ̂ + 6𝒋 ̂ + 8𝒌 ̂ (𝒃𝟏) ⃗ × (𝒃𝟐) ⃗ = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@7& −6&1@1&−2&1)| = 𝑖 ̂ [(−6 × 1)−(−2×1)] − 𝑗 ̂ [(7×1)−(1×1)] + 𝑘 ̂ [(7×−2)−(1×−6)] = 𝑖 ̂ [−6+2] − 𝑗 ̂ [7−1] + 𝑘 ̂ [−14+6] = −4𝒊 ̂ − 6𝒋 ̂ – 8𝒌 ̂ Magnitude of ((𝑏1) ⃗ × (𝑏2) ⃗) = √((−4)2+(−6)2+(−8)2) |(𝒃𝟏) ⃗ × (𝒃𝟐) ⃗ | = √(16+36+64) = √𝟏𝟏𝟔 Also, ((𝒃𝟏) ⃗ × (𝒃𝟐) ⃗) . ((𝒂𝟐) ⃗ – (𝒂𝟏) ⃗) = ("−4" 𝑖 ̂" − 6" 𝑗 ̂" – 8" 𝑘 ̂).(4𝑖 ̂ + 6𝑗 ̂ + 8𝑘 ̂) = –4 × 4 + (–6) × 6 + (–8) × 8 = –16 – 36 – 64 = – 116 So, Shortest distance = |(((𝑏_1 ) ⃗ × (𝑏_2 ) ⃗ ).((𝑎_2 ) ⃗ − (𝑎_1 ) ⃗ ))/|(𝑏_1 ) ⃗ × (𝑏_2 ) ⃗ | | = |( −116)/√116| = √𝟏𝟏𝟔 Therefore, shortest distance between the given two lines is (3√2)/2.