Questions

Let ${L}_{1}$ and ${L}_{2}$ denotes the lines $\overline{r}=\overline{i}+\lambda \left(-\overline{i}+2\overline{j}+2\overline{k}\right),\lambda \in R$ and $\overline{r}=\mu \left(2\overline{i}-\overline{j}+2\overline{k}\right),\mu \in R$ respectively. If ${L}_{3}$ is a line which is perpendicular to both ${L}_{1}$ and ${L}_{2}$ and cuts both of them, which of the following options discribes ${L}_{3}$?

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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

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detailed solution

Correct option is A

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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A line $l$ passing through the origin is perpendicular to the line ${l}_{1}\text{\hspace{0.17em}}:\text{\hspace{0.17em}}\left(3+t\right)\text{\hspace{0.17em}}\stackrel{^}{i}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(-1+\text{\hspace{0.17em}}2t\right)\text{\hspace{0.17em}}\stackrel{^}{j}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(4+\text{\hspace{0.17em}}2t\right)\text{\hspace{0.17em}}\stackrel{^}{k},-\infty
${l}_{2}\text{\hspace{0.17em}}:\text{\hspace{0.17em}}\left(3+2s\right)\text{\hspace{0.17em}}\stackrel{^}{i}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(3+\text{\hspace{0.17em}}2s\right)\text{\hspace{0.17em}}\stackrel{^}{j}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(2+s\right)\text{\hspace{0.17em}}\stackrel{^}{k}\text{\hspace{0.17em}},-\infty
Then the coordinate (s) of the point (s) on ${l}_{2}$ at a distance of  $\sqrt{17}$ from the point of intersection of $l$ and ${l}_{1}$is