The shortest distance (in units) between the lines L_1:(x-1)/2=(y-2)/3=(z+4)/6 and L_2:(x-3)/2=(3-y)/(-3)=(z+5)/6 is: Option 1: 293/7 Option 2: √293/7 Option 3: √293 Option 4: √493

Answer: √293/7. Solution: Rewriting L_2 as (x-3)/2=(y-3)/3=(z+5)/6 shows both lines share direction d=(2,3,6), so they are parallel. With a_2-a_1=(2,1,-1), the cross product is (9,-14,4) of magnitude √293. Shortest distance =√293/(, d, )=√293/7.

Geometry

3D Geometry

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The shortest distance (in units) between the lines $L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$ and $L_2:\frac{x-3}{2}=\frac{3-y}{-3}=\frac{z+5}{6}$ is:

$\frac{293}{7}$

$\frac{\sqrt{293}}{7}$

$\sqrt{293}$

$\sqrt{493}$

✅ Correct Option: 2

Rewriting $L_2$ as $\frac{x-3}{2}=\frac{y-3}{3}=\frac{z+5}{6}$ shows both lines share direction $\vec d=(2,3,6)$ , so they are parallel.

With $\vec{a_2}-\vec{a_1}=(2,1,-1)$ , the cross product is $(9,-14,4)$ of magnitude $\sqrt{293}$ .

Shortest distance $=\frac{\sqrt{293}}{|\vec d|}=\frac{\sqrt{293}}{7}$ .

2023: 30 May Shift 3

If the equation of a floor of a room is given by $x + y - z + 4 = 0$ and the equation of roof is given by $x + y - z + 5 = 0$ . Then, the height of the room is :

2026: 12 May Shift 2

If angle between the lines $\dfrac{x-1}{\propto} = \dfrac{y+2}{-5} = \dfrac{z+\frac{7}{3}}{\beta}$ and $x=z, y=0$ is $\dfrac{\pi}{4}$ , then

2026: 29 May Shift 2

If $\lambda$ and $\mu$ are parameters, then

Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	$\vec{r}=2\hat{i}+\hat{j}-3\hat{k}+\lambda(\hat{i}-\hat{j}+2\hat{k})$ parallel to the line	I. $\vec{r}=(\hat{j}+\hat{k})+\mu(-2\hat{i}+\hat{j}-2\hat{k})$
B.	$\vec{r}=\hat{i}+\hat{j}+\lambda(\hat{i}+\hat{j}-2\hat{k})$ perpendicular to the line	II. $\vec{r}=(2\hat{i}+\hat{k})+\mu(-2\hat{i}+\hat{j})$
C.	$\frac{x}{-2}=\frac{y-1}{1}=\frac{z-1}{-2}$ in vector form	III. $\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(2\hat{i}-2\hat{j}+4\hat{k})$
D.	$\frac{2-x}{2}=y, z=1$ in vector form	IV. $\frac{x-1}{2}=\frac{y+1}{4}=\frac{z}{3}$

Choose the correct answer from the options given below:

2026: 29 May Shift 1

A line in space is uniquely determined if

(A) It passes through a given point and has a given direction.

(B) It passes through two given points.

(D) It has a given direction.

Choose the correct answer from the options given below:

2025: 2 June Shift 1

If the direction ratios of two lines are $a, b, c$ and $(b-c), (c-a), (a-b)$ respectively, then the angle between these lines is:

2025: 22 May Shift 2

Match List-I with List-II

List-I	List-II
Equation of line	Information
(A) $\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(\hat{j} - 2\hat{k})$	(I) Direction ratios are 2, 4, -1
(B) $\frac{2-x}{1} = \frac{2y+1}{4}$ , $z = 2$	(II) Perpendicular to $2\hat{i} - \hat{j} + \hat{k}$
(C) $\frac{x}{1} = \frac{y-3}{2} = \frac{3-4z}{2}$	(III) Passing through the point $(3, -2, 1)$
(D) $\vec{r} = (3\hat{i} + 2\hat{j} + \hat{k}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$	(IV) Direction ratios are -1, 2, 0

Choose the correct answer from the options given below:

The shortest distance (in units) between the lines L1:x−12=y−23=z+46L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}L1​:2x−1​=3y−2​=6z+4​ and L2:x−32=3−y−3=z+56L_2:\frac{x-3}{2}=\frac{3-y}{-3}=\frac{z+5}{6}L2​:2x−3​=−33−y​=6z+5​ is:

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CUET Mathematics Past Year Questions (Topic-Wise)

The shortest distance (in units) between the lines $L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}$ and $L_2:\frac{x-3}{2}=\frac{3-y}{-3}=\frac{z+5}{6}$ is: