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Distance between two parallel lines

This type of question, asked 3 times: May 2022 S1, 2 May 2023 S2, 20 Apr 2025 S1. All 3 questions below, oldest first.

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  1. Question 1

    May 2022 S1

    111
    The distance between parallel lines x−12=y−2−2=z−31\frac{x-1}{2} = \frac{y-2}{-2} = \frac{z-3}{1} and x2=y−2=z1\frac{x}{2} = \frac{y}{-2} = \frac{z}{1} is

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  2. Question 2

    2 May 2023 S2

    109
    Distance between the parallel lines x3=y−1−2=z1\frac{x}{3} = \frac{y-1}{-2} = \frac{z}{1} and x+43=y−3−2=z+21\frac{x+4}{3} = \frac{y-3}{-2} = \frac{z+2}{1} is

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  3. Question 3

    20 Apr 2025 S1

    109
    If the shortest distance between the lines x−k2=y−43=z−34\frac{x- k}{2}=\frac{y- 4}{3}=\frac{z - 3}{4} and x−24=y−46=z−78\frac{x- 2}{4}=\frac{y- 4}{6}=\frac{z - 7}{8} is 1329\frac{13}{\sqrt{29}}, then k=k =

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