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Two lines intersect: find the unknown

This type of question, asked 13 times: 10 May 2023 S1, 10 May 2023 S2, 11 May 2023 S1, 15 May 2023 S2, 11 May 2024 S2, 14 May 2024 S2, 23 Apr 2025 S1, 11 Apr 2026 S1, 11 Apr 2026 S2, 15 Apr 2026 S2, 17 Apr 2026 S1, 12 May 2026 S1, 19 May 2026 S2. All 13 questions below, oldest first.

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  1. Question 1

    10 May 2023 S1

    122
    If the lines x−12=y−23=z−14\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-1}{4} and x−3−1=y−k2=z1\frac{x-3}{-1}=\frac{y-k}{2}=\frac{z}{1} intersect, then k is equal to

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  2. Question 2

    10 May 2023 S2

    126
    If the line x−12=y+13=z−14\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} and x−31=y−k2=z1\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1} intersect, then the value of k is

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  3. Question 3

    11 May 2023 S1

    124
    If the lines x+1−10=y+k−1=z−41\frac{x+1}{-10} = \frac{y+k}{-1} = \frac{z-4}{1} and x+10−1=y+1−3=z−14\frac{x+10}{-1} = \frac{y+1}{-3} = \frac{z-1}{4} intersect each other, then the value of k is

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  4. Question 4

    15 May 2023 S2

    127
    If the lines x−12=y+23=z−14\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-1}{4} and x−31=y−k2=z1\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1} intersect, then kk has the value

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  5. Question 5

    11 May 2024 S2

    104
    If the lines x−12=y+13=z−14\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} and x−31=y−k2=z1\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1} intersect, then the value of k is:

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  6. Question 6

    14 May 2024 S2

    137
    If the lines x−k2=y+13=z−14\frac{x-k}{2}=\frac{y+1}{3}=\frac{z-1}{4} and x−31=y−9/22=z1\frac{x-3}{1}=\frac{y-9/2}{2}=\frac{z}{1} intersect, then the value of kk is

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  7. Question 7

    23 Apr 2025 S1

    138
    If the lines x−12=y+13=z−14\frac{x- 1}{2}=\frac{y+ 1}{3}=\frac{z- 1}{4} and x−31=y−k2=z1\frac{x- 3}{1}=\frac{y- k}{2}=\frac{z}{1}. intersect, then the value of k is

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  8. Question 8

    11 Apr 2026 S1

    94
    If the lines x−12=y+13=z−14\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \dfrac{z - 1}{4} and x−21=y+m2=z−21\dfrac{x - 2}{1} = \dfrac{y + m}{2} = \dfrac{z - 2}{1} intersect each other, then the value of mm is....

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  9. Question 9

    11 Apr 2026 S2

    95
    If the lines x−12=y+13=z−14\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \dfrac{z - 1}{4} and x−31=y−c2=z1\dfrac{x - 3}{1} = \dfrac{y - c}{2} = \dfrac{z}{1} intersect, then the radius of circle x2+y2−4x+10y+c=0x^2 + y^2 - 4x + 10y + c = 0 is

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  10. Question 10

    15 Apr 2026 S2

    93
    If the lines r⃗=(i^+mj^+3k^)+λ(2i^+3j^+4k^)\vec{r} = (\hat{i} + m\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k}) and r⃗=(4i^+j^)+μ(5i^+mj^+k^)\vec{r} = (4\hat{i} + \hat{j}) + \mu(5\hat{i} + m\hat{j} + \hat{k}) intersect each other, then m =

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  11. Question 11

    17 Apr 2026 S1

    95
    If the lines r⃗=i^+j^−k^+λ(qi^−2j^+k^)\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda(q\hat{i} - 2\hat{j} + \hat{k}) and r⃗=pi^−3j^+2k^+μ(i^−2j^+2k^)\vec{r} = p\hat{i} - 3\hat{j} + 2\hat{k} + \mu(\hat{i} - 2\hat{j} + 2\hat{k}) intersect each other and qi^−2j^+k^q\hat{i} - 2\hat{j} + \hat{k} is collinear to 4i^−4j^+2k^4\hat{i} - 4\hat{j} + 2\hat{k}, then the values of pp and qq are

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  12. Question 12

    12 May 2026 S1

    96
    If the lines rˉ=(i^−j^+2k^)+λ(i^+2k^)\bar{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} + 2\hat{k}) and x−p2=y+2=z+1\dfrac{x - p}{2} = y + 2 = z + 1 intersect, then the value of pp is ...

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  13. Question 13

    19 May 2026 S2

    94
    If the lines x−11=y−12=z−13\dfrac{x - 1}{1} = \dfrac{y - 1}{2} = \dfrac{z - 1}{3} and x−31=y−k2=z−51\dfrac{x - 3}{1} = \dfrac{y - k}{2} = \dfrac{z - 5}{1} are intersect at the point PP, then the distance (in units) of the point PP from the origin O(0,0,0)O(0, 0, 0) is ...

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