If the shortest distance between the lines x + 6 2 = y - 6 3 = z - 6 4 and x - 3 = y - 2 6 4 = z + 2 6 5 is 6 , then the square of the sum of all possible values of is ________ . [2023]

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Solution

Q.
If the shortest distance between the lines $\frac{x + \sqrt{6}}{2} = \frac{y - \sqrt{6}}{3} = \frac{z - \sqrt{6}}{4}$ and $\frac{x - λ}{3} = \frac{y - 2 \sqrt{6}}{4} = \frac{z + 2 \sqrt{6}}{5}$ is $6$ , then the square of the sum of all possible values of $λ$ is ________ . [2023]

Ans.
(384)

$Given, shortest distance between the lines$

$\frac{x + \sqrt{6}}{2} = \frac{y - 2 \sqrt{6}}{3} = \frac{z - \sqrt{6}}{4}$

$and \frac{x - λ}{3} = \frac{y - 2 \sqrt{6}}{4} = \frac{z + 2 \sqrt{6}}{5} is 6 .$

$Vector along line of shortest distance$

$=$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{matrix} |$ $= \hat{i} (15 - 16) - \hat{j} (10 - 12) + \hat{k} (8 - 9)$

$= - \hat{i} + 2 \hat{j} - \hat{k} (its magnitude is \sqrt{6})$

$Now, \frac{1}{\sqrt{6}}$ $| \begin{matrix} λ + \sqrt{6} & \sqrt{6} & - 3 \sqrt{6} \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{matrix} |$ $= \pm 6$ $\Rightarrow λ = - 2 \sqrt{6}, 10 \sqrt{6}$

$So, square of sum of these values: {(10 \sqrt{6} - 2 \sqrt{6})}^{2} = {(8 \sqrt{6})}^{2} = 384$

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