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Shortest distance between two lines is given: find the unknown

This type of question, asked 2 times: 14 May 2024 S1, 19 Apr 2025 S2. All 2 questions below, oldest first.

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  1. Question 1

    14 May 2024 S1

    128
    If the shortest distance between the lines x−12=y−23=z−3λ\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{\lambda} and x−21=y−44=z−55\dfrac{x-2}{1}=\dfrac{y-4}{4}=\dfrac{z-5}{5} is 13\dfrac{1}{\sqrt{3}}, then sum of possible values of λ\lambda is

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  2. Question 2

    19 Apr 2025 S2

    149
    If the shortest distance between the lines r‾1=αi^+2j^+2k^+λ(i^−2j^+2k^),λ∈R,α>0{\overline{r}}_{1}=\alpha\widehat{i}+ 2\widehat{j}+ 2\widehat{k}+\lambda(\widehat{i}- 2\widehat{j}+ 2\widehat{k}),\lambda\in\mathbb{R},\alpha> 0 and rˉ2=−4i^−k^+μ(3i^−2j^−2k^),μ∈R{\bar{r}}_{2}= - 4\widehat{i}-\widehat{k}+\mu(3\widehat{i}- 2\widehat{j}- 2\widehat{k}),\mu\in R, is 9 , then the value of α\alpha is

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