The line L given by 2 x − 2 = b y − 1 = c z + 1 passes through the point (1, 2, 3). Another line K is parallel to line L and has the equation a x + 2 = 2 y − 3 = d z + 4 . Then the shortes

Question

The line L given by 2 x − 2 ​ = b y − 1 ​ = c z + 1 ​ passes through the point (1, 2, 3). Another line K is parallel to line L and has the equation a x + 2 ​ = 2 y − 3 ​ = d z + 4 ​ . Then the shortest distance between line L and K is

TestHub · Accepted Answer

(1,2,3) lie on L ​ ∴ − 2 1 ​ = b 1 ​ = c 4 ​ ​ ∴ b = − 2 , c = − 8 ​ Line L and k are parallel ​ a 2 ​ = 2 b ​ = d c ​ ⇒ a 2 ​ = − 1 = − d 8 ​ ​ ∴ a = − 2 , d = 8 ​ \begin{center} \end{center} ​ 2 x − 2 ​ = − 2 y − 1 ​ = − 8 z + 1 ​ ​ − 2 x + 2 ​ = 2 y − 3 ​ = 8 z + 4 ​ ​ cos θ = ​ 72 ​ 29 ​ 8 + 4 − 24 ​ ​ = 29 ​ 2 ​ ​ ∴ sin θ = 29 ​ 27 ​ ​

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