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Q.

Let L1 and L2 denotes the lines r¯=i¯+λ(−i¯+2j¯+2k¯),λ∈R and r¯=μ(2i¯−j¯+2k¯),μ∈R respectively. If L3 is a line which is perpendicular to both L1 and L2 and cuts both of them, which of the following options discribes L3?

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a

r¯=13(2i¯+k¯)+t(2i¯+2j¯−k¯),t∈R

b

r¯=29(2i¯+j¯+2k¯)+t(2i¯+2j¯−k¯),t∈R

c

r¯=t(2i¯+2j¯−k¯),t∈R

d

r¯=29(4i¯+j¯+k¯)+t(2i¯+2j¯−k¯),t∈R

answer is A.

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Detailed Solution

Point on L1 and L2 are respectivelyA(1−λ,2λ,2λ) and B(2μ,−μ,2μ)AB¯=(2μ+λ−1)i^+(−μ−2λ)j^+(2μ−2λ)k^vector along their shortest distance =2i^+2j^−k^Hence 2μ+λ−12=−μ−2λ2=2μ−2λ−1⇒λ=19 and μ=29

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