If the shortest distance between the lines x – 4 1 = y + 1 2 = z – 3 and x – 2 = y + 1 4 = z – 2 – 5 is 6 5 , then the sum of all possible values of is : [2024]

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Solution

Q.
If the shortest distance between the lines $\frac{x - 4}{1} = \frac{y + 1}{2} = \frac{z}{- 3}$ and $\frac{x - λ}{2} = \frac{y + 1}{4} = \frac{z - 2}{- 5}$ is $\frac{6}{\sqrt{5}}$ , then the sum of all possible values of $λ$ is : [2024]

1 5

2 7

3 8

4 10

Ans.
(3)

Here $x_{1} = 4, y_{1} = - 1, z_{1} = 0$ and $x_{2} = λ, y_{2} = - 1, z_{2} = 2$

$a_{1} = 1, b_{1} = 2, c_{1} = - 3$ and $a_{2} = 2, b_{2} = 4, c_{2} = - 5$

$∵ \frac{6}{\sqrt{5}} =$ $| \frac{| \begin{matrix} λ - 4 & 0 & 2 \\ 1 & 2 & - 3 \\ 2 & 4 & - 5 \end{matrix} |}{\sqrt{{(- 10 + 12)}^{2} + {(- 6 + 5)}^{2} + {(4 - 4)}^{2}}} |$

$\Rightarrow \frac{6}{\sqrt{5}} =$ $| \frac{2 λ - 8}{\sqrt{5}} |$ $\Rightarrow λ = 7 or 1$ .

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