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Two lines are coplanar: find the unknown

This type of question, asked 7 times: 4 May 2023 S1, 11 May 2023 S2, 19 Apr 2025 S2, 25 Apr 2025 S2, 15 Apr 2026 S2, 16 Apr 2026 S1, 17 Apr 2026 S1. All 7 questions below, oldest first.

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  1. Question 1

    4 May 2023 S1

    102
    The lines x−21=y−31=z−4−k\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k} and x−1k=y−42=z−51\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1} are coplanar if

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  2. Question 2

    11 May 2023 S2

    127
    If for some α∈R\alpha\in\mathbb{R}, the lines L1:x+12=y−2−1=z−11L_1:\frac{x+1}{2}=\frac{y-2}{-1}=\frac{z-1}{1} and L2:x+2α=y+15−α=z+11L_2:\frac{x+2}{\alpha}=\frac{y+1}{5-\alpha}=\frac{z+1}{1} are coplanar, then the line L2L_2 passes through the point

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  3. Question 3

    19 Apr 2025 S2

    142
    The lines x−31=y−21=z−5−k\frac{x- 3}{1}=\frac{y- 2}{1}=\frac{z- 5}{-k} and x−4k=y−31=z−32\frac{x- 4}{k}=\frac{y- 3}{1}=\frac{z - 3}{2} are coplanar, hence k=k =

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  4. Question 4

    25 Apr 2025 S2

    119
    If the lines x=ay−1=z−2x=ay- 1 =z- 2 and x=3y−2=bz−2(ab≠0)x= 3y- 2 = bz - 2(ab \neq 0) are coplanar, then

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  5. Question 5

    15 Apr 2026 S2

    94
    If the lines x=−1+s,y=3−λs,z=1+λsx = -1 + s, y = 3 - \lambda s, z = 1 + \lambda s and x=t2,y=1+t,z=2−tx = \dfrac{t}{2}, y = 1 + t, z = 2 - t with parameters s and t, are coplanar, then λ=\lambda =

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  6. Question 6

    16 Apr 2026 S1

    90
    The lines x−21=y−31=z−4−k\dfrac{x - 2}{1} = \dfrac{y - 3}{1} = \dfrac{z - 4}{-k} and x−1k=y−42=z−51\dfrac{x - 1}{k} = \dfrac{y - 4}{2} = \dfrac{z - 5}{1} are coplanar if

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  7. Question 7

    17 Apr 2026 S1

    94
    If for some m∈Rm \in \mathbb{R} the lines L1:x+1m=y−m−1=z−11L_1 : \dfrac{x + 1}{m} = \dfrac{y - m}{-1} = \dfrac{z - 1}{1} and L2:x+2−4=y+19=z+11L_2 : \dfrac{x + 2}{-4} = \dfrac{y + 1}{9} = \dfrac{z + 1}{1} are coplanar, then line L1L_1 passes through the point

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