Shortest Distance Between Two Skew Lines

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JEE Main 2012

If the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect, then $k$ is equal to:

- 1

(B)

\frac{2}{9}

(C)

\frac{9}{2}

(D)

0

JEE Main 2008

LEVELJEE Main

If the straight lines $\frac{x - 1}{k} = \frac{y - 2}{2} = \frac{z - 3}{3}$ and $\frac{x - 2}{3} = \frac{y - 3}{k} = \frac{z - 1}{2}$ intersect at a point, then the integer $k$ is equal to

(A)

- 5

(B)

5

(C)

2

(D)

- 2

JEE Main 2003

LEVELJEE Main

The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{- k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}$ are coplanar if

(A)

k = 3

- 2

(B)

k = 0

- 1

(C)

k = 1

- 1

(D)

k = 0

- 3

JEE Advanced 2004S

LEVELJEE Main

If the lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect, then the value of $k$ is

(A)

3/2

(B)

9/2

(C)

- 2/9

(D)

- 3/2

JEE Advanced 2008

LEVELJEE Advanced

Comprehension Passage

Consider the lines

L_{1} : \frac{x + 1}{3} = \frac{y + 2}{1} = \frac{z + 1}{2}

and

L_{2} : \frac{x - 2}{1} = \frac{y + 2}{2} = \frac{z - 3}{3}

Question 1:

The unit vector perpendicular to both $L_{1}$ and $L_{2}$ is

(A)

\frac{- i ^ + 7 j ^ + 7 k ^}{99}

(B)

\frac{- i ^ - 7 j ^ + 5 k ^}{5 3}

(C)

\frac{- i ^ + 7 j ^ + 5 k ^}{5 3}

(D)

\frac{7 i ^ - 7 j ^ - k ^}{99}

Question 2:

The shortest distance between $L_{1}$ and $L_{2}$ is

(A)

(B)

\frac{17}{3}

(C)

\frac{41}{5 3}

(D)

\frac{17}{5 3}

Question 3:

The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines $L_{1}$ and $L_{2}$ is

(A)

\frac{2}{75}

(B)

\frac{7}{75}

(C)

\frac{13}{75}

(D)

\frac{23}{75}

JEE Main 2020 (8 January Shift 1)

LEVELJEE Main

The shortest distance between the lines $\frac{x - 3}{3} = \frac{y - 8}{- 1} = \frac{z - 3}{1}$ and $\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}$ is :

(A)

230

(B)

\frac{7}{2} 30

(C)

3

(D)

330

JEE Main 2020 - 8 Jan (Morning)

LEVELJEE Main

The shortest distance between the lines $\frac{x - 3}{3} = \frac{y - 8}{- 1} = \frac{z - 8}{1}$ and $\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}$ is:

(A)

330

(B)

230

(C)

30

(D)

430

JEE Main 2020 - 6 Sep (Morning)

LEVELJEE Advanced

The shortest distance between the lines $\frac{x - 1}{0} = \frac{y + 1}{- 1} = \frac{z}{1}$ and $x + y + z + 1 = 0, 2 x - y + z + 3 = 0$ is:

(A)

1/ 2

(B)

1

(C)

1/ 3

(D)

1/2

JEE Main 2021 (20 July Shift 1)

LEVELJEE Main

If the shortest distance between the lines $r_{1} = α \hat{i} + 2 \hat{j} + 2 \hat{k} + λ (\hat{i} - 2 \hat{j} + 2 \hat{k}), λ \in R, α > 0$ and $r_{2} = - 4 \hat{i} - \hat{k} + μ (3 \hat{i} - 2 \hat{j} - 2 \hat{k}), μ \in R$ is 9, then $α$ is equal to ___

JEE Main 2021 (22 July Shift 1)

LEVELJEE Main

If the shortest distance between the straight lines $3 (x - 1) = 6 (y - 2) = 2 (z - 1)$ and $4 (x - 2) = 2 (y - λ) = (z - 3), λ \in R$ is $\frac{1}{38}$ , then the integral value of $λ$ is equal to :

JEE Main 2021 (25 July Shift 2)

LEVELJEE Main

If the lines $\frac{x - k}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$ and $\frac{x + 1}{3} = \frac{y + 2}{2} = \frac{z + 3}{1}$ are co-planar, then the value of $k$ is ___

JEE Main 2021 (24 February Shift 2)

LEVELJEE Main

Let $λ$ be an integer. If the shortest distance between the lines $x - λ = 2 y - 1 = - 2 z$ and $x = y + 2 λ = z - λ$ is $\frac{7}{2 2}$ , then the value of $∣ λ ∣$ is

JEE Main 2022 (24 June Shift 2)

LEVELJEE Main

If the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{λ}$ and $\frac{x - 2}{1} = \frac{y - 4}{4} = \frac{z - 5}{5}$ is $\frac{1}{3}$ , then the sum of all possible values of $λ$ is :

JEE Main 2022 (25 June Shift 2)

LEVELJEE Advanced

Let $l_{1}$ be the line in $x y$ -plane with $x$ and $y$ intercepts $\frac{1}{8}$ and $\frac{1}{4 2}$ respectively, and $l_{2}$ be the line in $z x$ -plane with $x$ and $z$ intercepts $- \frac{1}{8}$ and $- \frac{1}{6 3}$ respectively. If $d$ is the shortest distance between the line $l_{1}$ and $l_{2}$ , then $d^{- 2}$ is equal to ____.

JEE Main 2022 (27 June Shift 2)

LEVELJEE Main

The shortest distance between the lines $\frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{- 1}$ and $\frac{x + 3}{2} = \frac{y - 6}{1} = \frac{z - 5}{3}$ is:

(A)

\frac{18}{5}

(B)

\frac{22}{3 5}

(C)

\frac{46}{3 5}

(D)

63

JEE Main 2022 (24 June Shift 1)

LEVELJEE Main

If the shortest distance between the line $r = (- \hat{i} + 3 \hat{k}) + λ (\hat{i} - a \hat{j})$ and $r = (- \hat{j} + 2 \hat{k}) + μ (\hat{i} - \hat{j} + \hat{k})$ is $\frac{2}{3}$ , then the integral value of $a$ is equal to

JEE Main 2022 (25 July Shift 1)

LEVELJEE Advanced

The line of shortest distance between the lines $\frac{x - 2}{0} = \frac{y - 1}{1} = \frac{z}{1}$ and $\frac{x - 3}{2} = \frac{y - 5}{2} = \frac{z - 1}{1}$ makes an angle of $cos^{- 1} \frac{2}{27}$ with the plane $P : a x - y - z = 0, (a > 0)$ . If the image of the point $(1, 1, - 5)$ in the plane $P$ is $(α, β, γ)$ , then $α + β - γ$ is equal to ______. (Enter 0 if no such plane exists)

JEE Main 2022 (25 July Shift 2)

LEVELJEE Main

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

(A)

229

(B)

(C)

\frac{37}{29}

(D)

\frac{29}{2}

JEE Main 2023 (06 April Shift 1)

LEVELJEE Main

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

(A)

\frac{12}{5}

(B)

125

(C)

\frac{12}{5 5}

(D)

\frac{12}{5}

JEE Main 2023 (24 January Shift 1)

LEVELJEE Main

The shortest distance between the lines $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 6}{2}$ and $\frac{x - 6}{3} = \frac{1 - y}{2} = \frac{z + 8}{0}$ is equal to ______.

JEE Main 2023 (24 January Shift 2)

LEVELJEE Main

If the shortest between the lines $\frac{x + 6}{2} = \frac{y - 6}{3} = \frac{z - 6}{4}$ and $\frac{x - λ}{3} = \frac{y - 2 6}{4} = \frac{z + 2 6}{5}$ is 6, then the square of sum of all possible values of $λ$ is

JEE Main 2023 (25 January Shift 2)

LEVELJEE Main

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

(A)

(B)

(C)

5/2

(D)

3/2

JEE Main 2023 (25 January Shift 2)

LEVELJEE Main

If the shortest distance between the line joining the points $(1, 2, 3)$ and $(2, 3, 4)$ and the line $\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 2}{0}$ is $α$ , then $28 α^{2}$ is equal to ______.

JEE Main 2023 (29 January Shift 2)

LEVELJEE Main

Shortest distance between the lines $\frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}$ and $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{- 3}$ is

(A)

23

(B)

43

(C)

33

(D)

53

JEE Main 2023 (30 January Shift 1)

LEVELJEE Main

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

(A)

(B)

19/3

(C)

19/3

(D)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line 2x−1​=3y+1​=4z−1​ and 1x−3​=2y−k​=1z​ intersect, then k is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the straight lines kx−1​=2y−2​=3z−3​ and 3x−2​=ky−3​=2z−1​ intersect at a point, then the integer k is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The lines 1x−2​=1y−3​=−kz−4​ and kx−1​=2y−4​=1z−5​ are coplanar if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 2x−1​=3y+1​=4z−1​ and 1x−3​=2y−k​=1z​ intersect, then the value of k is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The unit vector perpendicular to both L1​ and L2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between L1​ and L2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines L1​ and L2​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between the lines 3x−3​=−1y−8​=1z−3​ and −3x+3​=2y+7​=4z−6​ is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between the lines 3x−3​=−1y−8​=1z−8​ and −3x+3​=2y+7​=4z−6​ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between the lines 0x−1​=−1y+1​=1z​ and x+y+z+1=0,2x−y+z+3=0 is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the shortest distance between the straight lines 3(x−1)=6(y−2)=2(z−1) and 4(x−2)=2(y−λ)=(z−3),λ∈R is 38​1​, then the integral value of λ is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the lines 1x−k​=2y−2​=3z−3​ and 3x+1​=2y+2​=1z+3​ are co-planar, then the value of k is ___

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let λ be an integer. If the shortest distance between the lines x−λ=2y−1=−2z and x=y+2λ=z−λ is 22​7​​, then the value of ∣λ∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the shortest distance between the lines 2x−1​=3y−2​=λz−3​ and 1x−2​=4y−4​=5z−5​ is 3​1​, then the sum of all possible values of λ is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between the lines 2x−3​=3y−2​=−1z−1​ and 2x+3​=1y−6​=3z−5​ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the shortest distance between the line r=(−i^+3k^)+λ(i^−aj^​) and r=(−j^​+2k^)+μ(i^−j^​+k^) is 32​​, then the integral value of a is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between the lines −6x+7​=7y−6​=z and 27−x​=y−2=z−6 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between the lines 3x−2​=2y+1​=2z−6​ and 3x−6​=21−y​=0z+8​ is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the shortest between the lines 2x+6​​=3y−6​​=4z−6​​ and 3x−λ​=4y−26​​=5z+26​​ is 6, then the square of sum of all possible values of λ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between the lines x+1=2y=−12z and x=y+2=6z−6 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the shortest distance between the line joining the points (1,2,3) and (2,3,4) and the line 2x−1​=−1y+1​=0z−2​ is α, then 28α2 is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Shortest distance between the lines 2x−1​=−7y+8​=5z−4​ and 2x−1​=1y−2​=−3z−6​ is

If the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect, then $k$ is equal to:

If the straight lines $\frac{x - 1}{k} = \frac{y - 2}{2} = \frac{z - 3}{3}$ and $\frac{x - 2}{3} = \frac{y - 3}{k} = \frac{z - 1}{2}$ intersect at a point, then the integer $k$ is equal to

The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{- k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}$ are coplanar if

If the lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect, then the value of $k$ is

The unit vector perpendicular to both $L_{1}$ and $L_{2}$ is

The shortest distance between $L_{1}$ and $L_{2}$ is

The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines $L_{1}$ and $L_{2}$ is

The shortest distance between the lines $\frac{x - 3}{3} = \frac{y - 8}{- 1} = \frac{z - 3}{1}$ and $\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}$ is :

The shortest distance between the lines $\frac{x - 3}{3} = \frac{y - 8}{- 1} = \frac{z - 8}{1}$ and $\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}$ is:

The shortest distance between the lines $\frac{x - 1}{0} = \frac{y + 1}{- 1} = \frac{z}{1}$ and $x + y + z + 1 = 0, 2 x - y + z + 3 = 0$ is:

If the shortest distance between the straight lines $3 (x - 1) = 6 (y - 2) = 2 (z - 1)$ and $4 (x - 2) = 2 (y - λ) = (z - 3), λ \in R$ is $\frac{1}{38}$ , then the integral value of $λ$ is equal to :

If the lines $\frac{x - k}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}$ and $\frac{x + 1}{3} = \frac{y + 2}{2} = \frac{z + 3}{1}$ are co-planar, then the value of $k$ is ___

Let $λ$ be an integer. If the shortest distance between the lines $x - λ = 2 y - 1 = - 2 z$ and $x = y + 2 λ = z - λ$ is $\frac{7}{2 2}$ , then the value of $∣ λ ∣$ is

If the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{λ}$ and $\frac{x - 2}{1} = \frac{y - 4}{4} = \frac{z - 5}{5}$ is $\frac{1}{3}$ , then the sum of all possible values of $λ$ is :

The shortest distance between the lines $\frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{- 1}$ and $\frac{x + 3}{2} = \frac{y - 6}{1} = \frac{z - 5}{3}$ is:

If the shortest distance between the line $r = (- \hat{i} + 3 \hat{k}) + λ (\hat{i} - a \hat{j})$ and $r = (- \hat{j} + 2 \hat{k}) + μ (\hat{i} - \hat{j} + \hat{k})$ is $\frac{2}{3}$ , then the integral value of $a$ is equal to

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

The shortest distance between the lines $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 6}{2}$ and $\frac{x - 6}{3} = \frac{1 - y}{2} = \frac{z + 8}{0}$ is equal to ______.

If the shortest between the lines $\frac{x + 6}{2} = \frac{y - 6}{3} = \frac{z - 6}{4}$ and $\frac{x - λ}{3} = \frac{y - 2 6}{4} = \frac{z + 2 6}{5}$ is 6, then the square of sum of all possible values of $λ$ is

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

If the shortest distance between the line joining the points $(1, 2, 3)$ and $(2, 3, 4)$ and the line $\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 2}{0}$ is $α$ , then $28 α^{2}$ is equal to ______.

Shortest distance between the lines $\frac{x - 1}{2} = \frac{y + 8}{- 7} = \frac{z - 4}{5}$ and $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{- 3}$ is