Let S be the set of all values of , for which the shortest distance between the lines x -0 = y - 3 4 = z + 6 1 and x + = y - 4 = z - 6 0 is 13. Then 8 is equal to [2023]

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Solution

Q.
Let S be the set of all values of $λ$ , for which the shortest distance between the lines $\frac{x - λ}{0} = \frac{y - 3}{4} = \frac{z + 6}{1}$ and $\frac{x + λ}{3} = \frac{y}{- 4} = \frac{z - 6}{0}$ is 13. Then $8$ $| \sum_{λ \in S} λ |$ is equal to [2023]

1 302

2 306

3 308

4 304

Ans.
(2)

$Shortest distance between lines$

$\frac{x - λ}{0} = \frac{y - 3}{4} = \frac{z + 6}{1}$ and $\frac{x + λ}{3} = \frac{y}{- 4} = \frac{z - 6}{0}$ is 13

So, $| \frac{| \begin{matrix} λ + λ & 3 - 0 & - 6 - 6 \\ 0 & 4 & 1 \\ 3 & - 4 & 0 \end{matrix} |}{\sqrt{{(0 + 4)}^{2} + {(3 - 0)}^{2} + {(0 - 12)}^{2}}} |$ $= 13$

$\Rightarrow$ $| \frac{| \begin{matrix} 2 λ & 3 & - 12 \\ 0 & 4 & 1 \\ 3 & - 4 & 0 \end{matrix} |}{13} |$ $= 13$

$\Rightarrow$ $| 2 λ (0 + 4) - 3 (- 3) - 12 (0 - 12) |$ $= 13 \times 13$

$\Rightarrow$ $| 8 λ + 9 + 144 |$ $= 169$ $\Rightarrow$ $| 8 λ + 153 |$ $= 169$

$\Rightarrow 8 λ + 153 = 169$ and $8 λ + 153 = - 169 \Rightarrow λ = 2, \frac{- 161}{4}$

$∴$ $8$ $| \sum_{λ \in S} λ |$ $= 8$ $| \frac{- 161}{4} + 2 |$ $= 8$ $| \frac{- 153}{4} |$ $= 153 \times 2 = 306$

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