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Shortest distance between two skew lines

This type of question, asked 33 times: 2021 Paper 2, 2021 Paper 6, 14 Jun 2022, 25 Jun 2022, 27 Jun 2022, 25 Jul 2022, 24 Jan 2023, 25 Jan 2023, 29 Jan 2023, 30 Jan 2023, 31 Jan 2023, 1 Feb 2023, 6 Apr 2023, 8 Apr 2023, 10 Apr 2023, 15 Apr 2023, 27 Jan 2024, 30 Jan 2024, 31 Jan 2024, 1 Feb 2024, 4 Apr 2024, 6 Apr 2024, 8 Apr 2024, 9 Apr 2024, 23 Jan 2025, 2 Apr 2025, 3 Apr 2025, 4 Apr 2025, 7 Apr 2025, 22 Jan 2026 S1, 24 Jan 2026 S2, 4 Apr 2026 S2, 6 Apr 2026 S2. All 38 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    85
    Let λ\lambda be an integer. If the shortest distance between the lines x−λ=2y−1=−2zx-\lambda= 2y- 1 = - 2z and x=y+2λ=z−λx=y+ 2\lambda=z-\lambda is 722\frac{\sqrt{7}}{2\sqrt{2}}, then the value of ∣λ∣|\lambda| is

  2. Question 2

    2021 Paper 6

    78
    If the foot of the perpendicular from point (4,3,8)(4,3,8) on the line L1:x−al=y−23=z−b4L_{1}:\frac{x - a}{l}=\frac{y - 2}{3}=\frac{z - b}{4}, l≠0l\neq 0 is (3,5,7)(3,5,7), then the shortest distance between the line L1L_{1} and line L2:x−23=y−44=z−55L_{2}:\frac{x - 2}{3}=\frac{y - 4}{4}=\frac{z - 5}{5} is equal to:

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  3. Question 3

    14 Jun 2022

    164
    If the shortest distance between the lines x−12=y−23=z−3λ\frac{x - 1}{2}=\frac{y - 2}{3}=\frac{z - 3}{\lambda} and x−21=y−44=z−55\frac{x - 2}{1}=\frac{y - 4}{4}=\frac{z - 5}{5} is 13\frac{1}{\sqrt{3}}, then the sum of all possible values of λ\lambda is:

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  4. Question 4

    14 Jun 2022

    90
    If the shortest distance between the line r→=(−i^+3k)+λ(i^−aj^)\overrightarrow{r}= ( -\widehat{i}+ 3k) +\lambda(\widehat{i}-a\widehat{j}) and r→=(−j^+2k)+μ(i^−j^+k)\overrightarrow{r}= ( -\widehat{j}+ 2k) +\mu(\widehat{i}-\widehat{j}+k) is 23\sqrt{\frac{2}{3}}, then the integral value of a is equal to

  5. Question 5

    25 Jun 2022

    180
    Let l1l_{1} be the line in xyxy-plane with xx and yy intercepts 18\frac{1}{8} and 142\frac{1}{4\sqrt{2}} respectively, and l2l_{2} be the line in zx-plane with xx and zz intercepts −18-\frac{1}{8} and −163-\frac{1}{6\sqrt{3}} respectively. If dd is the shortest distance between the line l1l_{1} and l2l_{2}, then d−2d^{- 2} is equal to

  6. Question 6

    27 Jun 2022

    164
    The shortest distance between the lines x−32=y−23=z−1−1\frac{x - 3}{2}=\frac{y - 2}{3}=\frac{z - 1}{- 1} and x+32=y−61=z−53\frac{x + 3}{2}=\frac{y - 6}{1}=\frac{z - 5}{3} is :

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  7. Question 7

    25 Jul 2022

    166
    The shortest distance between the lines x+7−6=y−67=z\frac{x + 7}{- 6}=\frac{y - 6}{7}= z and 7−x2=y−2=z−6\frac{7 - x}{2}= y - 2 = z - 6 is

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  8. Question 8

    24 Jan 2023

    83
    The shortest distance between the lines x−23=y+12=z−62\frac{x- 2}{3}=\frac{y+ 1}{2}=\frac{z- 6}{2} and x−63=1−y2=z+80\frac{x- 6}{3}=\frac{1 -y}{2}=\frac{z+ 8}{0} is equal to

  9. Question 9

    24 Jan 2023

    171
    If the shortest distance between the lines x+62=y−63=z−64\frac{x +\sqrt{6}}{2}=\frac{y -\sqrt{6}}{3}=\frac{z -\sqrt{6}}{4} and x−λ3=y−264=z+265\frac{x - \lambda}{3}=\frac{y - 2\sqrt{6}}{4}=\frac{z + 2\sqrt{6}}{5} is 6 , then the square of sum of all possible values of λ\lambda is

  10. Question 10

    25 Jan 2023

    170
    The shortest distance between the lines x+1=2y=−12zx + 1 = 2y = - 12z and x=y+2=6z−6x = y + 2 = 6z - 6 is

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  11. Question 11

    25 Jan 2023

    176
    If the shortest distance between the line joining the points (1,2,3)(1,2,3) and (2,3,4)(2,3,4), and the line x−12=y+1−1=z−20\frac{x - 1}{2}=\frac{y + 1}{- 1}=\frac{z - 2}{0} is a, then 28a228a^{2} is equal to

  12. Question 12

    29 Jan 2023

    157
    The shortest distance between the lines x−12=y−21=z−6−3\frac{x - 1}{2}=\frac{y - 2}{1}=\frac{z - 6}{- 3} and x−12=y+8−7=z−45\frac{x - 1}{2}=\frac{y + 8}{- 7}=\frac{z - 4}{5}

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  13. Question 13

    30 Jan 2023

    73
    The line l1l_{1} passes through the point (2,6,2)(2,6,2) and is perpendicular to the plane 2x+y−2z=102x + y - 2z = 10. Then the shortest distance between the line l1l_{1} and the line x+12=y+4−3=z2\frac{x + 1}{2}=\frac{y + 4}{- 3}=\frac{z}{2} is:

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  14. Question 14

    31 Jan 2023

    71
    Let the shortest distance between the lines L:x−5−2=y−λ0=z+λ1,λ≥0L:\frac{x - 5}{- 2}=\frac{y - \lambda}{0}=\frac{z + \lambda}{1},\lambda \geq 0 and L1:x+1=y−1=4−zL_{1}:x + 1 = y - 1 = 4 - z be 262\sqrt{6}. If (α,β,γ)(\alpha,\beta,\gamma) lies on LL, then which of the following is NOT possible?

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  15. Question 15

    1 Feb 2023

    72
    The shortest distance between the lines x−51=y−22=z−4−3\frac{x - 5}{1}=\frac{y - 2}{2}=\frac{z - 4}{- 3} and x+31=y+54=z−1−5\frac{x + 3}{1}=\frac{y + 5}{4}=\frac{z - 1}{- 5} is

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  16. Question 16

    6 Apr 2023

    78
    One vertex of a rectangular parallelopiped is at the origin OO and the lengths of its edges along x,yx,y and zz axes are 3, 4 and 5 units respectively. Let PP be the vertex (3,4,5)(3,4,5). Then the shortest distance between the diagonal OP and an edge parallel to zz axis, not passing through OO or PP is:

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  17. Question 17

    8 Apr 2023

    76
    The shortest distance between the lines x−44=y+25=z+33\frac{x - 4}{4}=\frac{y + 2}{5}=\frac{z + 3}{3} and x−13=y−34=z−42\frac{x - 1}{3}=\frac{y - 3}{4}=\frac{z - 4}{2} is

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  18. Question 18

    10 Apr 2023

    77
    The shortest distance between the lines x+21=y−2=z−52\frac{x + 2}{1}=\frac{y}{- 2}=\frac{z - 5}{2} and x−41=y−12=z+30\frac{x - 4}{1}=\frac{y - 1}{2}=\frac{z + 3}{0} is

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  19. Question 19

    15 Apr 2023

    62
    Let SS be the set of all values of λ\lambda, for which the shortest distance between the lines x−λ0=y−34=z+61\frac{x - \lambda}{0}=\frac{y - 3}{4}=\frac{z + 6}{1} and x+λ3=y−4=z−60\frac{x + \lambda}{3}=\frac{y}{- 4}=\frac{z - 6}{0} is 13 . Then 8∣∑λ∈Sλ∣8\left| \sum_{\lambda \in S} \lambda \right| is equal to

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  20. Question 20

    27 Jan 2024

    68
    If the shortest distance between the lines x−41=y+12=z−3\frac{x - 4}{1}=\frac{y + 1}{2}=\frac{z}{- 3} and x−λ2=y+14=z−2−5\frac{x - \lambda}{2}=\frac{y + 1}{4}=\frac{z - 2}{- 5} is 65\frac{6}{\sqrt{5}}, then the sum of all possible values of λ\lambda is :

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  21. Question 21

    30 Jan 2024

    82
    If d1d_{1} is the shortest distance between the lines x+1=2y=−12z,x=y+2=6z−6x+ 1 = 2y= - 12z,x=y+ 2 = 6z- 6 and d2d_{2} is the shortest distance between the lines x−12=y+8−7=z−45,x−12=y−21=z−6−3\frac{x- 1}{2}=\frac{y+ 8}{- 7}=\frac{z- 4}{5},\frac{x- 1}{2}=\frac{y- 2}{1}=\frac{z- 6}{- 3}, then the value of 323d1d2\frac{32\sqrt{3}d_{1}}{d_{2}} is :

  22. Question 22

    31 Jan 2024

    170
    The shortest distance between lines L1L_{1} and L2L_{2}, where L1:x−12=y+1−3=z+42L_{1}:\frac{x - 1}{2}=\frac{y + 1}{- 3}=\frac{z + 4}{2} and L2L_{2} is the line passing through the points A(−4,4,3)⋅B(−1,6,3)A( - 4,4,3) \cdot B( - 1,6,3) and perpendicular to the line x−3−2=y3=z−11\frac{x - 3}{- 2}=\frac{y}{3}=\frac{z - 1}{1}, is

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  23. Question 23

    1 Feb 2024

    80
    If the shortest distance between the lines x−λ−2=y−21=z−11\frac{x - \lambda}{- 2}=\frac{y - 2}{1}=\frac{z - 1}{1} and x−31=y−1−2=z−21\frac{x -\sqrt{3}}{1}=\frac{y - 1}{- 2}=\frac{z - 2}{1} is 1 , then the sum of all possible values of λ\lambda is :

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  24. Question 24

    4 Apr 2024

    84
    If the shortest distance between the lines x+22=y+33=z−54\frac{x + 2}{2}=\frac{y + 3}{3}=\frac{z - 5}{4} and x−31=y−2−3=z+42\frac{x - 3}{1}=\frac{y - 2}{- 3}=\frac{z + 4}{2} is 3835k\frac{38}{3\sqrt{5}}k and  ∫0k[x2]dx=α−α,\ \int_{0}^{k} \left\lbrack x^{2} \right\rbrack dx = \alpha -\sqrt{\alpha}, where  [x]\ \lbrack x\rbrack denotes the greatest integer function, then 6α36\alpha^{3} is equal to

  25. Question 25

    6 Apr 2024

    177
    If the shortest distance between the lines x−λ3=y−2−1=z−11\frac{x-\lambda}{3}=\frac{y- 2}{- 1}=\frac{z- 1}{1} and x+2−3=y+52=z−44\frac{x+ 2}{- 3}=\frac{y+ 5}{2}=\frac{z- 4}{4} is 4430\frac{44}{\sqrt{30}}, then the largest possible value of ∣λ∣|\lambda| is equal to

  26. Question 26

    6 Apr 2024

    69
    The shortest distance between the lines x−32=y+15−7=z−95\frac{x - 3}{2}=\frac{y + 15}{- 7}=\frac{z - 9}{5} and x+12=y−11=z−9−3\frac{x + 1}{2}=\frac{y - 1}{1}=\frac{z - 9}{- 3} is

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  27. Question 27

    8 Apr 2024

    69
    If the shortest distance between the lines
    L1:r→=(2+λ)i^+(1−3λ)j^+(3+4λ)k^,λ∈RL_{1}:\overrightarrow{r}= (2 +\lambda)\widehat{i}+ (1 - 3\lambda)\widehat{j}+ (3 + 4\lambda)\widehat{k},\lambda\in\mathbb{R}
    L2:r→=2(1+μ)i^+3(1+μ)j^+(5+μ)k^,μ∈RL_{2}:\overrightarrow{r}= 2(1 +\mu)\widehat{i}+ 3(1 +\mu)\widehat{j}+ (5 +\mu)\widehat{k},\mu\in\mathbb{R}
    is mn\frac{m}{\sqrt{n}}, where gcd(m,n)=1gcd(m,n) = 1, then the value of m+nm+n equals.

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  28. Question 28

    8 Apr 2024

    158
    If the shortest distance between the lines x−λ2=y−43=z−34\frac{x - \lambda}{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 a value of λ\lambda is :

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  29. Question 29

    9 Apr 2024

    75
    The shortest distance between the line x−34=y+7−11=z−15\frac{x- 3}{4}=\frac{y+ 7}{- 11}=\frac{z- 1}{5} and x−53=y−9−6=z+21\frac{x- 5}{3}=\frac{y- 9}{- 6}=\frac{z+ 2}{1} is :

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  30. Question 30

    23 Jan 2025

    143
    If the square of the shortest distance between the lines x−21=y−12=z+3−3\frac{x - 2}{1}=\frac{y - 1}{2}=\frac{z + 3}{- 3} and x+12=y+34=z+5−5\frac{x + 1}{2}=\frac{y + 3}{4}=\frac{z + 5}{- 5} is mn\frac{m}{n}, where m,nm,n are coprime numbers, then m+nm + n is equal to:

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  31. Question 31

    2 Apr 2025

    131
    The line LlL_{l} is parallel to the vector a→=−3i^+2j^+4k^\overrightarrow{a}= - 3\widehat{i}+ 2\widehat{j}+ 4\widehat{k} and passes through the point (7,6,2)(7,6,2) and the line L2L_{2} is parallel to the vector b→=2i^+j^+3k^\overrightarrow{b}= 2\widehat{i}+\widehat{j}+ 3\widehat{k} and passes through the point (5, 3, 4). The shortest distance between the lines L1L_{1} and L2L_{2} is :

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  32. Question 32

    3 Apr 2025

    70
    Line L1L_{1} passes through the point (1,2,3)(1,2,3) and is parallel to zz-axis. Line L2L_{2} passes through the point (λ,5,6)(\lambda,5,6) and is parallel to yy-axis. Let for λ=λ1,λ2,λ2<λ1\lambda=\lambda_{1},\lambda_{2},\lambda_{2}<\lambda_{1}, the shortest distance between the two lines be 3 . Then the square of the distance of the point (λ1,λ2,7)\left( \lambda_{1},\lambda_{2},7 \right) from the line L1L_{1} is.

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  33. Question 33

    4 Apr 2025

    56
    Let the shortest distance between the lines x−33=y−α−1=z−31\frac{x- 3}{3}=\frac{y-\alpha}{- 1}=\frac{z- 3}{1} and x+3−3=y+72=z−β4\frac{x+ 3}{- 3}=\frac{y+ 7}{2}=\frac{z-\beta}{4} be 3303\sqrt{30}. Then the positive value of 5α+β5\alpha+\beta is.

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  34. Question 34

    7 Apr 2025

    52
    If the shortest distance between the lines x−12=y−23=z−34\frac{x- 1}{2}=\frac{y- 2}{3}=\frac{z- 3}{4} and x1=yα=z−51\frac{x}{1}=\frac{y}{\alpha}=\frac{z- 5}{1} is 56\frac{5}{\sqrt{6}}, then the sum of all possible values of α\alpha is.

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  35. Question 35

    22 Jan 2026 S1

    56
    Let P(α,β,γ)P(\alpha,\beta,\gamma) be the point on the line x−12=y+1−3=z\frac{x - 1}{2}=\frac{y + 1}{- 3}= z at a distance 4144\sqrt{14} from the point (1,−1,0)(1, - 1,0) and nearer to the origin. Then the shortest distance, between the lines x−α1=y−β2=z−γ3\frac{x - \alpha}{1}=\frac{y - \beta}{2}=\frac{z - \gamma}{3} and x+52=y−101=z−31\frac{x + 5}{2}=\frac{y - 10}{1}=\frac{z - 3}{1} , is equal to

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  36. Question 36

    24 Jan 2026 S2

    61
    The sum of all values of α\alpha, for which the shortest distance between the lines  x+1α=y−2−1=z−4−α\ \ \frac{x + 1}{\alpha}=\frac{y - 2}{- 1}=\frac{z - 4}{- \alpha} and xα=y−12=z−12α\frac{x}{\alpha}=\frac{y - 1}{2}=\frac{z - 1}{2\alpha} is 2\sqrt{2}, is

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  37. Question 37

    4 Apr 2026 S2

    63
    The shortest distance between the lines r→=(13i^+2j^+83k^)+λ(2i^−5j^+6k^)\overrightarrow{r}=\left( \frac{1}{3}\widehat{i}+ 2\widehat{j}+\frac{8}{3}\widehat{k} \right)+\lambda(2\widehat{i}- 5\widehat{j}+ 6\widehat{k}) and r→=(−23i^−13k^)+μ(j^−k^),λ,μ∈R\overrightarrow{r}=\left( -\frac{2}{3}\widehat{i}-\frac{1}{3}\widehat{k} \right)+\mu(\widehat{j}-\widehat{k}),\lambda,\mu\in\mathbb{R}, is :

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  38. Question 38

    6 Apr 2026 S2

    64
    The shortest distance between the lines x−41=y−32=z−2−3\frac{x - 4}{1}=\frac{y - 3}{2}=\frac{z - 2}{- 3} and x+22=y−64=z−5−5\frac{x + 2}{2}=\frac{y - 6}{4}=\frac{z - 5}{- 5} is:

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