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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

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a
73,73,53
b
(-1, -1, 0)
c
(1,1,1)
d
79,79,83

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detailed solution

Correct option is B

The line l is perpendicular to l1 and l2. Hence the direction ratios of  l are given by the vectori^j^k^122221=− 2i^  +  3j^ − 2k^Let P ≡ (2r, − 3r, 2r)As it lies in l1, we have 2r− 31=−3r+ 12=2r−42Thus we have r=1 and P≡  (2, − 3, 2)A point on l2 is then (3+ 2s,  3 + 2s, 2 + s)(3 + 2s − 2)2 + (3 + 2s + 3)2 + (2 + s− 2)2  = 17⇒  9s2 + 28s + 20 = 0 given s = − 2, − 10/9Thus the points are  (−1,   1, 0) and 79, 79 ,89

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 ${L}_{3}$?