Core Syllabus Module

Three Dimensional Geometry

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2023

LEVELJEE Advanced

Let $P$ be the plane $3 x + 2 y + 3 z = 16$ and let $S = {α \hat{i} + β \hat{j} + γ \hat{k} : α^{2} + β^{2} + γ^{2} = 1 and the distance of (α, β, γ) from the plane P is \frac{7}{2}}$ . Let $u, v$ and $w$ be three distinct vectors in $S$ such that $∣ u - v ∣ = ∣ v - w ∣ = ∣ w - u ∣$ . Let $V$ be the volume of the parallelepiped determined by vectors $u, v$ and $w$ . Then the value of $\frac{80}{3} V$ is

JEE Main 2021 (March) (16 March Shift 1)

LEVELJEE Advanced

Let $P$ be a plane $l x + m y + n z = 0$ containing the line, $\frac{x - 1}{1} = \frac{y + 4}{2} = \frac{z + 2}{3}$ . If plane $P$ divides the line segment $A B$ joining points $A (- 3, - 6, 1)$ and $B (2, 4, - 3)$ in ratio $k : 1$ then the value of $k$ is equal to :

JEE Main 2022 (26 July Shift 2)

LEVELJEE Advanced

The plane passing through the line $L : ℓ x - y + 3 (1 - ℓ) z = 1, x + 2 y - z = 2$ and perpendicular to the plane $3 x + 2 y + z = 6$ is $3 x - 8 y + 7 z = 4$ . If $θ$ is the acute angle between the line $L$ and the y-axis, then $415 cos^{2} θ$ 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 (29 June Shift 2)

LEVELJEE Advanced

Let $Q$ be the mirror image of the point $P (1, 2, 1)$ with respect to the plane $x + 2 y + 2 z = 16$ . Let $T$ be a plane passing through the point $Q$ and contains the line $r = - \hat{k} + λ (\hat{i} + \hat{j} + 2 \hat{k}), λ \in R$ . Then, which of the following points lies on $T$ ?

(A)

(2, 1, 0)

(B)

(1, 2, 1)

(C)

(1, 2, 2)

(D)

(1, 3, 2)

JEE Advanced 2006

LEVELJEE Advanced

Match the following :

List-I

(P)

Two rays

x + y = ∣ a ∣

and

a x - y = 1

intersects each other in the first quadrant in the interval

a \in (a_{0}, \infty)

, the value of

a_{0}

(Q)

Point

(α, β, γ)

lies on the plane

x + y + z = 2

. Let

a = α \hat{i} + β \hat{j} + γ \hat{k}

\hat{k} \times (\hat{k} \times a) = 0

, then

γ =

(R)

\int_{0}^{1} (1 - y^{2}) d y + \int_{1}^{0} (y^{2} - 1) d y

(S)

sin A sin B sin C + cos A cos B = 1

, then the value of

sin C =

List-II

\int_{0}^{1} 1 - x d x + \int_{- 1}^{0} 1 + x d x

(4)

JEE Main 2021 (26 Aug Shift 1)

LEVELJEE Advanced

Let the line $L$ be the projection of the line $\frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 4}{2}$ in the plane $x - 2 y - z = 3$ . If $d$ is the distance of the point $(0, 0, 6)$ from $L$ , then $d^{2}$ is equal to .

JEE Main 2022 (28 July Shift 1)

LEVELJEE Advanced

The foot of the perpendicular from a point on the circle $x^{2} + y^{2} = 1, z = 0$ to the plane $2 x + 3 y + z = 6$ lies on which one of the following curves ?

(A)

(6 x + 5 y - 12)^{2} + 4 (3 x + 7 y - 8)^{2} = 1, z = 6 - 2 x - 3 y

(B)

(5 x + 6 y - 12)^{2} + 4 (3 x + 5 y - 9)^{2} = 1, z = 6 - 2 x - 3 y

(C)

(6 x + 5 y - 14)^{2} + 9 (3 x + 5 y - 7)^{2} = 1, z = 6 - 2 x - 3 y

(D)

(5 x + 6 y - 14)^{2} + 9 (3 x + 7 y - 8)^{2} = 1, z = 6 - 2 x - 3 y

JEE Advanced 2024

LEVELJEE Advanced

Let $R^{3}$ denote the three dimensional space. Take two points $P = (1, 2, 3)$ and $Q = (4, 2, 7)$ . Let $dist (X, Y)$ denote the distance between two points $X$ and $Y$ in $R^{3}$ . Let $S = {X \in R^{3} : (dist (X, P))^{2} - (dist (X, Q))^{2} = 50}$ and $T = {Y \in R^{3} : (dist (Y, Q))^{2} - (dist (Y, P))^{2} = 50}$ . Then which of the following statements is(are) TRUE?

* Multiple Correct Options

(A)

There is a triangle whose area is 1 and all of whose vertices are from S.

(B)

There are two distinct points L and M in T such that each point on the line segments LM is also in T.

(C)

There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T.

(D)

There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T.

JEE Advanced 2023

LEVELJEE Advanced

Let $ℓ_{1}$ and $ℓ_{2}$ be the lines $r_{1} = λ (\hat{i} + \hat{j} + \hat{k})$ and $r_{2} = (\hat{j} - \hat{k}) + μ (\hat{i} + \hat{k})$ , respectively. Let $X$ be the set of all the planes $H$ that contain the line $ℓ_{1}$ . For a plane $H$ , let $d (H)$ denote the smallest possible distance between the points of $ℓ_{2}$ and $H$ . Let $H_{0}$ be plane in $X$ for which $d (H_{0})$ is the maximum value of $d (H)$ as $H$ varies over all planes in $X$ . Match each entry in List-I to the correct entries in List-II.

List-I

(P)

(P) The value of

d (H_{0})

(Q)

(Q) The distance of the point

(0, 1, 2)

from

H_{0}

(R)

(R) The distance of origin from

H_{0}

(S)

(S) The distance of origin from the point of intersection of planes

y = z, x = 1

and

H_{0}

List-II

(1)

3

(2)

\frac{1}{3}

(3)

(3) 0

(4)

2

(5)

\frac{1}{2}

JEE Main 2022 (27 July Shift 1)

LEVELJEE Advanced

Let the line $\frac{x - 3}{7} = \frac{y - 2}{- 1} = \frac{z - 3}{- 4}$ intersect the plane containing the lines $\frac{x - 4}{1} = \frac{y + 1}{- 2} = \frac{z}{1}$ and $4 a x - y + 5 z - 7 a = 0 = 2 x - 5 y - z - 3, a \in R$ at the point $P (α, β, γ)$ . Then the value of $α + β + γ$ equals ______.

JEE Main 2023 (12 April Shift 1)

LEVELJEE Advanced

Let the plane $x + 3 y - 2 z + 6 = 0$ meet the co-ordinate axes at the points $A, B, C$ . If the orthocenter of the triangle $A B C$ is $(α, β, \frac{6}{7})$ , then $98 (α + β)^{2}$ is equal to __________.

JEE Main 2024 (04 Apr Shift 1)

LEVELJEE Advanced

If the shortest distance between the lines $\frac{x + 2}{2} = \frac{y + 3}{3} = \frac{z - 5}{4}$ and $\frac{x - 3}{1} = \frac{y - 2}{- 3} = \frac{z + 4}{2}$ is $\frac{38}{3 5} k$ , and $\int_{0}^{k} [x^{2}] d x = α - α$ , where $[x]$ denotes the greatest integer function, then $6 α^{3}$ is equal to

JEE Advanced 2022

LEVELJEE Advanced

Let $P_{1}$ and $P_{2}$ be two planes given by $P_{1} : 10 x + 15 y + 12 z - 60 = 0$ , $P_{2} : - 2 x + 5 y + 4 z - 20 = 0$ . Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_{1}$ and $P_{2}$ ?

* Multiple Correct Options

(A)

\frac{x - 1}{0} = \frac{y - 1}{0} = \frac{z - 1}{5}

(B)

\frac{x - 6}{- 5} = \frac{y}{2} = \frac{z}{3}

(C)

\frac{x}{- 2} = \frac{y - 4}{5} = \frac{z}{4}

(D)

\frac{x}{1} = \frac{y - 4}{- 2} = \frac{z}{3}

JEE Main 2024 (09 Apr Shift 1)

LEVELJEE Advanced

Let the line $L$ intersect the lines $x - 2 = - y = z - 1, 2 (x + 1) = 2 (y - 1) = z + 1$ and be parallel to the line $\frac{x - 2}{3} = \frac{y - 1}{1} = \frac{z - 2}{2}$ . Then which of the following points lies on L?

(A)

(- \frac{1}{3}, 1, - 1)

(B)

(- \frac{1}{3}, - 1, 1)

(C)

(- \frac{1}{3}, 1, 1)

(D)

(- \frac{1}{3}, - 1, - 1)

JEE Main 2026 (23 January Shift 2)

LEVELJEE Advanced

If the image of the point $P (a, 2, a)$ in the line $\frac{x}{2} = \frac{y + a}{1} = \frac{z}{1}$ is $Q$ and the image of $Q$ in the line $\frac{x - 2 b}{2} = \frac{y - a}{1} = \frac{z + 2 b}{- 5}$ is $P$ , then $a + b$ is equal to .

JEE Main 2025 April

LEVELJEE Advanced

Let a line passing through the point $(4, 1, 0)$ intersect the line $L_{1} : \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ at the point $A (α, β, γ)$ and the line $L_{2} : x - 6 = y = - z + 4$ at the point $B (a, b, c)$ . Then $1 α a 0 β b 1 γ c$ is equal to

(A)

8

(B)

16

(C)

12

(D)

6

JEE Advanced 1996

LEVELJEE Advanced

The position vectors of the vertices $A, B$ and $C$ of a tetrahedron $A B C D$ are $\hat{i} + \hat{j} + \hat{k}, \hat{i}$ and $3 \hat{i}$ , respectively. The altitude from vertex $D$ to the opposite face $A B C$ meets the median line through $A$ of the triangle $A B C$ at a point $E$ . If the length of the side $A D$ is $4$ and the volume of the tetrahedron is $\frac{2 2}{3}$ , find the position vector of the point $E$ for all its possible positions.

JEE Advanced 2019

LEVELJEE Advanced

Let $L_{1}$ and $L_{2}$ denotes the lines $r = \hat{i} + λ (- \hat{i} + 2 \hat{j} + 2 \hat{k}), λ \in R$ and $r = μ (2 \hat{i} - \hat{j} + 2 \hat{k}), μ \in R$ respectively. If $L_{3}$ is a line which is perpendicular to both $L_{1}$ and $L_{2}$ and cuts both of them, then which of the following options describe(s) $L_{3}$ ?

* Multiple Correct Options

(A)

r = \frac{1}{3} (2 \hat{i} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in R

(B)

r = \frac{2}{9} (2 \hat{i} - \hat{j} + 2 \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in R

(C)

r = t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in R

(D)

r = \frac{2}{9} (4 \hat{i} + \hat{j} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in R

JEE Advanced 2016

LEVELJEE Advanced

Consider a pyramid $O P QR S$ located in the first octant ( $x \geq 0, y \geq 0, z \geq 0$ ) with $O$ as origin, and $O P$ and $O R$ along the x-axis and the y-axis, respectively. The base $O P QR$ of the pyramid is a square with $O P = 3$ . The point $S$ is directly above the mid-point, $T$ of diagonal $O Q$ such that $T S = 3$ . Then

* Multiple Correct Options

(A)

the acute angle between

O Q

and

O S

\frac{π}{3}

(B)

the equation of the plane containing the triangle

O QS

x - y = 0

(C)

the length of the perpendicular from

P

to the plane containing the triangle

O QS

\frac{3}{2}

(D)

the perpendicular distance from

O

to the straight line containing

R S

\frac{15}{2}

Micro-Topic Strategy Matrix

Direction Cosines and Direction Ratios

Equation of a Line in Space

Angle Between Two Lines

Shortest Distance Between Two Skew Lines

Equation of a Plane

Angle Between Two Planes

Intersection of a Line and a Plane

Chapter Elite-20 Challenge Zone

Match the following :

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the line L be the projection of the line 2x−1​=1y−3​=2z−4​ in the plane x−2y−z=3. If d is the distance of the point (0,0,6) from L, then d2 is equal to .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The foot of the perpendicular from a point on the circle x2+y2=1,z=0 to the plane 2x+3y+z=6 lies on which one of the following curves ?

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the plane x+3y−2z+6=0 meet the co-ordinate axes at the points A,B,C. If the orthocenter of the triangle ABC is (α,β,76​), then 98(α+β)2 is equal to __________.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the image of the point P(a,2,a) in the line 2x​=1y+a​=1z​ is Q and the image of Q in the line 2x−2b​=1y−a​=−5z+2b​ is P, then a+b is equal to .

Let the line $L$ be the projection of the line $\frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 4}{2}$ in the plane $x - 2 y - z = 3$ . If $d$ is the distance of the point $(0, 0, 6)$ from $L$ , then $d^{2}$ is equal to .

The foot of the perpendicular from a point on the circle $x^{2} + y^{2} = 1, z = 0$ to the plane $2 x + 3 y + z = 6$ lies on which one of the following curves ?

Let the plane $x + 3 y - 2 z + 6 = 0$ meet the co-ordinate axes at the points $A, B, C$ . If the orthocenter of the triangle $A B C$ is $(α, β, \frac{6}{7})$ , then $98 (α + β)^{2}$ is equal to __________.

If the image of the point $P (a, 2, a)$ in the line $\frac{x}{2} = \frac{y + a}{1} = \frac{z}{1}$ is $Q$ and the image of $Q$ in the line $\frac{x - 2 b}{2} = \frac{y - a}{1} = \frac{z + 2 b}{- 5}$ is $P$ , then $a + b$ is equal to .