3D Geometry

32 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q14·MCQ·Easy–moderate
Open
Foot of perpendicular to line
Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $\vec r = (-\hat i + 3\hat j + \hat k) + \lambda(2\hat i + 3\hat j - \hat k)$ . Then the length of the projection of the vector $\alpha\hat i + \beta\hat j + \gamma\hat k$ on the vector $6\hat i + 2\hat j + 3\hat k$ is:
1. (1) $\dfrac{15}{7}$
2. (2)4
3. (3) $\dfrac{18}{7}$
4. (4)3
MathsJEE Main 21 Jan 2026, Shift 2·Q7·MCQ·Moderate
Open
Distance of point from line
Let the line $L$ pass through the point $(-3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of $L$ from the point $(-2, r, 1)$ is $\sqrt{\dfrac{14}{3}}$ , then the sum of all possible values of $r$ is:
1. (1)12
2. (2)16
3. (3)6
4. (4)10
MathsJEE Main 21 Jan 2026, Shift 2·Q8·MCQ·Moderate
Open
Line intersecting two lines
Let the line $L_1$ be parallel to the vector $-3\hat i + 2\hat j + 4\hat k$ and pass through the point $(2, 6, 7)$ and the line $L_2$ be parallel to the vector $2\hat i + \hat j + 3\hat k$ and pass through the point $(4, 3, 5)$ . If the line $L_3$ is parallel to the vector $-3\hat i + 5\hat j + 16\hat k$ and intersects the lines $L_1$ and $L_2$ at the points $C$ and $D$ , respectively, then $\lvert\overrightarrow{CD}\rvert^2$ is equal to:
1. (1)171
2. (2)290
3. (3)312
4. (4)89
MathsJEE Main 22 Jan 2026, Shift 1·Q6·MCQ·Moderate
Open
Shortest distance between skew lines
Let $P(\alpha, \beta, \gamma)$ be the point on the line $\dfrac{x-1}{2} = \dfrac{y+1}{-3} = z$ at a distance $4\sqrt{14}$ from the point $(1, -1, 0)$ and nearer to the origin. Then the shortest distance, between the lines $\frac{x-\alpha}{1} = \frac{y-\beta}{2} = \frac{z-\gamma}{3} \quad\text{and}\quad \frac{x+5}{2} = \frac{y-10}{1} = \frac{z-3}{1},$ is equal to
1. (1) $7\sqrt{\dfrac54}$
2. (2) $4\sqrt{\dfrac75}$
3. (3) $4\sqrt{\dfrac57}$
4. (4) $2\sqrt{\dfrac74}$
MathsJEE Main 22 Jan 2026, Shift 1·Q11·MCQ·Moderate
Open
Image of point in line
If the image of the point $P(1, 2, a)$ in the line $\dfrac{x-6}{3} = \dfrac{y-7}{2} = \dfrac{7-z}{2}$ is $Q(5, b, c)$ , then $a^2 + b^2 + c^2$ is equal to
1. (1)293
2. (2)264
3. (3)298
4. (4)283
MathsJEE Main 22 Jan 2026, Shift 2·Q10·MCQ·Moderate
Open
Points on a line at distance
Let $L$ be the line $\dfrac{x+1}{2} = \dfrac{y+1}{3} = \dfrac{z+3}{6}$ and let $S$ be the set of all points $(a, b, c)$ on $L$ , whose distance from the line $\dfrac{x+1}{2} = \dfrac{y+1}{3} = \dfrac{z-9}{0}$ along the line $L$ is 7. Then $\displaystyle\sum_{(a,b,c)\in S}(a + b + c)$ is equal to:
1. (1)34
2. (2)28
3. (3)40
4. (4)6
MathsJEE Main 23 Jan 2026, Shift 1·Q15·MCQ·Moderate
Open
Foot of perpendicular to a line
The vertices $B$ and $C$ of a triangle $ABC$ lie on the line $\dfrac{x}{1} = \dfrac{1 - y}{-2} = \dfrac{z - 2}{3}$ . The coordinates of $A$ and $B$ are $(1, 6, 3)$ and $(4, 9, \alpha)$ respectively and $C$ is at a distance of 10 units from $B$ . The area (in sq. units) of $\Delta ABC$ is:
1. (1) $5\sqrt{13}$
2. (2) $15\sqrt{13}$
3. (3) $20\sqrt{13}$
4. (4) $10\sqrt{13}$
MathsJEE Main 23 Jan 2026, Shift 1·Q20·MCQ·Moderate
Open
Angle between lines from direction cosines
Let the direction cosines of two lines satisfy the equations: $4\ell + m - n = 0$ and $2mn + 10n\ell + 3\ell m = 0$ . Then the cosine of the acute angle between these lines is:
1. (1) $\dfrac{10}{\sqrt{38}}$
2. (2) $\dfrac{20}{3\sqrt{38}}$
3. (3) $\dfrac{10}{7\sqrt{38}}$
4. (4) $\dfrac{10}{3\sqrt{38}}$
MathsJEE Main 23 Jan 2026, Shift 2·Q22·Numerical·Tough
Open
Image of a point in line
If the image of the point $P(a, 2, a)$ in the line $\dfrac{x}{2} = \dfrac{y + a}{1} = \dfrac{z}{1}$ is $Q$ and the image of $Q$ in the line $\dfrac{x - 2b}{2} = \dfrac{y - a}{1} = \dfrac{z + 2b}{-5}$ is $P$ , then $a + b$ is equal to .
We corrected a printing error in the original paper
Source printed “and the of image of Q in the line”.
MathsJEE Main 24 Jan 2026, Shift 1·Q3·MCQ·Moderate
Open
Intersecting lines and distances
Let the lines $L_1 : \vec r = \hat i + 2\hat j + 3\hat k + \lambda\left(2\hat i + 3\hat j + 4\hat k\right)$ , $\lambda \in \mathbb{R}$ and $L_2 : \vec r = \left(4\hat i + \hat j\right) + \mu\left(5\hat i + 2\hat j + \hat k\right)$ , $\mu \in \mathbb{R}$ , intersect at the point $R$ . Let $P$ and $Q$ be the points lying on lines $L_1$ and $L_2$ , respectively, such that $\left\lvert\overrightarrow{PR}\right\rvert = \sqrt{29}$ and $\left\lvert\overrightarrow{PQ}\right\rvert = \sqrt{\dfrac{47}{3}}$ . If the point $P$ lies in the first octant, then $27(QR)^2$ is equal to
1. (1)340
2. (2)360
3. (3)320
4. (4)348
MathsJEE Main 24 Jan 2026, Shift 1·Q22·Numerical·Moderate
Open
Line perpendicular to two lines
Let a line $L$ passing through the point $P(1, 1, 1)$ be perpendicular to the lines $\dfrac{x - 4}{4} = \dfrac{y - 1}{1} = \dfrac{z - 1}{1} \quad\text{and}\quad \dfrac{x - 17}{1} = \dfrac{y - 71}{1} = \dfrac{z}{0}.$ Let the line $L$ intersect the $yz$ -plane at the point $Q$ . Another line parallel to $L$ and passing through the point $S(1, 0, -1)$ intersects the $yz$ -plane at the point $R$ . Then the square of the area of the parallelogram $PQRS$ is equal to .
MathsJEE Main 24 Jan 2026, Shift 2·Q11·MCQ·Moderate
Open
Shortest distance between skew lines
The sum of all values of $\alpha$ , for which the shortest distance between the lines $\frac{x + 1}{\alpha} = \frac{y - 2}{-1} = \frac{z - 4}{-\alpha} \quad\text{and}\quad \frac{x}{\alpha} = \frac{y - 1}{2} = \frac{z - 1}{2\alpha}$ is $\sqrt2$ , is
1. (1)8
2. (2) $-6$
3. (3)6
4. (4) $-8$
MathsJEE Main 28 Jan 2026, Shift 1·Q18·MCQ·Moderate
Open
Distance along a line
If the distances of the point $(1, 2, a)$ from the line $\dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1}$ along the lines $L_1 : \frac{x-1}{3} = \frac{y-2}{4} = \frac{z-a}{b} \quad\text{and}\quad L_2 : \frac{x-1}{1} = \frac{y-2}{4} = \frac{z-a}{c}$ are equal, then $a + b + c$ is equal to
1. (1)7
2. (2)5
3. (3)6
4. (4)4
MathsJEE Main 28 Jan 2026, Shift 2·Q16·MCQ·Moderate
Open
Image of point in a line
Let $Q(a, b, c)$ be the image of the point $P(3, 2, 1)$ in the line $\dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1}$ . Then the distance of $Q$ from the line $\dfrac{x-9}{3} = \dfrac{y-9}{2} = \dfrac{z-5}{-2}$ is
1. (1)6
2. (2)8
3. (3)7
4. (4)5
MathsJEE Main 28 Jan 2026, Shift 2·Q24·Numerical·Moderate
Open
Distance along a given direction
If the distance of the point $P(43, \alpha, \beta)$ , $\beta < 0$ , from the line $\vec r = 4\hat i - \hat k + \mu(2\hat i + 3\hat k)$ , $\mu \in \mathbb{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$ , then $\alpha^2 + \beta^2$ is equal to .
MathsJEE Main 2 Apr 2026, Shift 1·Q13·MCQ·Moderate
Open
Intersection of two lines
If the point of intersection of the lines $\dfrac{x+1}{3} = \dfrac{y+a}{5} = \dfrac{z+b+1}{7}$ and $\dfrac{x-2}{1} = \dfrac{y-b}{4} = \dfrac{z-2a}{7}$ lies on xy-plane, then the value of $a + b$ is:
1. (1)2
2. (2)5
3. (3)7
4. (4)9
MathsJEE Main 2 Apr 2026, Shift 1·Q15·MCQ·Moderate
Open
Foot of perpendicular to line
Let a line $L$ passing through the point $(1, 1, 1)$ be perpendicular to both the vectors $2\hat i + 2\hat j + \hat k$ and $\hat i + 2\hat j + 2\hat k$ . If $P(a, b, c)$ is the foot of perpendicular from the origin on the line $L$ , then the value of $34(a + b + c)$ is:
1. (1)50
2. (2)80
3. (3)100
4. (4)120
We corrected a printing error in the original paper
Source printed the second vector as “ $\hat i + 2\hat j + \hat k$ ”; corrected to the intended “ $\hat i + 2\hat j + 2\hat k$ ” (the source's own solution uses it; as printed $34(a+b+c) = 95.2$ , no option).
MathsJEE Main 2 Apr 2026, Shift 2·Q15·MCQ·Moderate
Open
Foot of perpendicular on line
Let the point $A$ be the foot of perpendicular drawn from the point $P(a, b, 0)$ on the line $\dfrac{x-1}{2} = \dfrac{y-2}{1} = \dfrac{z-\alpha}{3}$ . If the midpoint of the line segment $PA$ is $\left(0, \dfrac34, \dfrac{-1}{4}\right)$ , then the value of $a^2 + b^2 + \alpha^2$ is equal to:
1. (1)1
2. (2)2
3. (3)6
4. (4)9
MathsJEE Main 4 Apr 2026, Shift 1·Q14·MCQ·Moderate
Open
Line intersecting two given lines
A line with direction ratios $1, -1, 2$ intersects the lines $\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z + 1}{3}$ and $\dfrac{x + 1}{-1} = \dfrac{y - 2}{1} = \dfrac{z}{4}$ at the points $P$ and $Q$ , respectively. If the length of the line segment $PQ$ is $\alpha$ , then $225\alpha^2$ is equal to:
1. (1)1024
2. (2)1014
3. (3)1104
4. (4)1204
MathsJEE Main 4 Apr 2026, Shift 1·Q15·MCQ·Moderate
Open
Distance measured along a line
The square of the distance of the point $(-2, -8, 6)$ from the line $\dfrac{x - 1}{1} = \dfrac{y - 1}{2} = \dfrac{z}{-1}$ along the line $\dfrac{x + 5}{1} = \dfrac{y + 5}{-1} = \dfrac{z}{2}$ is equal to:
1. (1)3
2. (2)6
3. (3)8
4. (4)12
MathsJEE Main 4 Apr 2026, Shift 2·Q13·MCQ·Easy–moderate
Open
Shortest distance between skew lines
The shortest distance between the lines $\vec r = \left(\frac13\hat i + 2\hat j + \frac83\hat k\right) + \lambda\left(2\hat i - 5\hat j + 6\hat k\right)$ and $\vec r = \left(-\dfrac23\hat i - \dfrac13\hat k\right) + \mu\left(\hat j - \hat k\right)$ , $\lambda, \mu \in \mathbb{R}$ , is:
1. (1) $\sqrt5$
2. (2)3
3. (3) $2\sqrt3$
4. (4) $\sqrt{15}$
MathsJEE Main 4 Apr 2026, Shift 2·Q14·MCQ·Moderate
Open
Image of point in line
If $\left(2\alpha + 1, \alpha^2 - 3\alpha, \dfrac{\alpha - 1}{2}\right)$ is the image of $(\alpha, 2\alpha, 1)$ in the line $\dfrac{x-2}{3} = \dfrac{y-1}{2} = \dfrac{z}{1}$ , then the possible value(s) of $\alpha$ is(are):
1. (1)Only 3
2. (2)Only 3 and $-1$
3. (3)Only 3, $\dfrac14$ and $-1$
4. (4)Only 3 and $\dfrac14$
MathsJEE Main 5 Apr 2026, Shift 1·Q13·MCQ·Easy–moderate
Open
Distance of point from line
The square of the distance of the point $P(5, 6, 7)$ from the line $\dfrac{x-2}{2} = \dfrac{y-5}{3} = \dfrac{z-2}{4}$ is equal to:
1. (1)3
2. (2)5
3. (3)6
4. (4)8
MathsJEE Main 5 Apr 2026, Shift 1·Q15·MCQ·Easy–moderate
Open
Intersection of two lines
The square of the distance of the point of intersection of the lines $\vec r = \left(\hat i + \hat j - \hat k\right) + \lambda(a\hat i - \hat j)$ , $a \neq 0$ and $\vec r = \left(4\hat i - \hat k\right) + \mu(2\hat i + a\hat k)$ from the origin is:
1. (1)5
2. (2)10
3. (3)17
4. (4)26
MathsJEE Main 5 Apr 2026, Shift 2·Q14·MCQ·Moderate
Open
Points on line at given distance
Let a triangle $PQR$ be such that $P$ and $Q$ lie on the line $\dfrac{x+3}{8} = \dfrac{y-4}{2} = \dfrac{z+1}{2}$ and are at a distance of 6 units from $R(1, 2, 3)$ . If $(\alpha, \beta, \gamma)$ is the centroid of $\Delta PQR$ , then $\alpha + \beta + \gamma$ is equal to:
1. (1)4
2. (2)5
3. (3)6
4. (4)8
MathsJEE Main 5 Apr 2026, Shift 2·Q15·MCQ·Moderate
Open
Image of point in line
If the distance of the point $(a, 2, 5)$ from the image of the point $(1, 2, 7)$ in the line $\dfrac{x}{1} = \dfrac{y-1}{1} = \dfrac{z-2}{2}$ is 4, then the sum of all possible values of $a$ is equal to:
1. (1)11
2. (2)9
3. (3)6
4. (4)4
MathsJEE Main 6 Apr 2026, Shift 1·Q15·MCQ·Tough
Open
Image of point in line
Let the image of the point $P(1, 6, a)$ in the line $L : \dfrac{x}{1} = \dfrac{y - 1}{2} = \dfrac{z - a + 1}{b}$ , $b > 0$ , be $\left(\dfrac{a}{3}, 0, a + c\right)$ . If $S(\alpha, \beta, \gamma)$ , $\alpha > 0$ , is the point on $L$ such that the distance of $S$ from the foot of perpendicular from the point $P$ on $L$ is $2\sqrt{14}$ , then $\alpha + \beta + \gamma$ is equal to:
1. (1)19
2. (2)20
3. (3)21
4. (4)22
MathsJEE Main 6 Apr 2026, Shift 1·Q16·MCQ·Easy–moderate
Open
Angle between two lines
Let a line $L$ be perpendicular to both the lines $L_1 : \dfrac{x + 1}{3} = \dfrac{y + 3}{5} = \dfrac{z + 5}{7} \quad\text{and}\quad L_2 : \dfrac{x - 2}{1} = \dfrac{y - 4}{4} = \dfrac{z - 6}{7}.$ If $\theta$ is the acute angle between the lines $L$ and $L_3 : \dfrac{x - \frac87}{2} = \dfrac{y - \frac47}{1} = \dfrac{z}{2},$ then $\tan\theta$ is equal to:
1. (1) $\dfrac32\sqrt2$
2. (2) $\dfrac52\sqrt2$
3. (3) $\dfrac53\sqrt2$
4. (4) $\dfrac43\sqrt2$
MathsJEE Main 6 Apr 2026, Shift 2·Q14·MCQ·Easy–moderate
Open
Shortest distance between skew lines
The shortest distance between the lines $\dfrac{x-4}{1} = \dfrac{y-3}{2} = \dfrac{z-2}{-3}$ and $\dfrac{x+2}{2} = \dfrac{y-6}{4} = \dfrac{z-5}{-5}$ is:
1. (1) $\dfrac{5\sqrt6}{6}$
2. (2) $2\sqrt5$
3. (3) $3\sqrt5$
4. (4) $4\sqrt5$
MathsJEE Main 6 Apr 2026, Shift 2·Q24·Numerical·Moderate
Open
Image of point in line
Let the image of the point $P(0, -5, 0)$ in the line $\dfrac{x-1}{2} = \dfrac{y}{1} = \dfrac{z+1}{-2}$ be the point $R$ and the image of the point $Q\left(0, \dfrac{-1}{2}, 0\right)$ in the line $\dfrac{x-1}{-1} = \dfrac{y+9}{4} = \dfrac{z+1}{1}$ be the point $S$ . Then the square of the area of the parallelogram $PQRS$ is .
MathsJEE Main 8 Apr 2026, Shift 2·Q15·MCQ·Moderate
Open
Distance between parallel lines
Let the foot of perpendicular from the point $(\lambda, 2, 3)$ on the line $\dfrac{x-4}{1} = \dfrac{y-9}{2} = \dfrac{z-5}{1}$ be the point $(1, \mu, 2)$ . Then the distance between the lines $\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z+4}{6}$ and $\dfrac{x-\lambda}{2} = \dfrac{y-\mu}{3} = \dfrac{z+5}{6}$ is equal to
1. (1) $\dfrac{12}{7}$
2. (2) $\dfrac{\sqrt{145}}{7}$
3. (3) $\dfrac{\sqrt{146}}{7}$
4. (4) $\dfrac{\sqrt{143}}{7}$
MathsJEE Main 8 Apr 2026, Shift 2·Q23·Numerical·Moderate
Open
Line perpendicular to two lines
Let a line $L_1$ pass through the origin and be perpendicular to the lines
$\begin{align*} L_2 &: \vec r = (3 + t)\hat i + (2t - 1)\hat j + (2t + 4)\hat k \quad\text{and}\\ L_3 &: \vec r = (3 + 2s)\hat i + (3 + 2s)\hat j + (2 + s)\hat k, \end{align*}$
$t, s \in \mathbb{R}$ . If $(a, b, c)$ , $a \in \mathbb{Z}$ , is the point on $L_3$ at a distance of $\sqrt{17}$ from the point of intersection of $L_1$ and $L_2$ , then $(a + b + c)^2$ is equal to .