Let L 1 and L 2 denotes the lines, r → = i ^ + λ - i ^ + 2 j ^ + 2 k ^ , λ ∈ R and r → = μ 2 i ^ - j ^ + 2 k ^ , μ ∈ 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 ?

r → = 2 9 2 i ^ - j ^ + 2 k ^ + t 2 i ^ + 2 j ^ - k ^ , t ∈ R

r → = t 2 i ^ + 2 j ^ - k ^ , t ∈ R

HARD

JEE Advanced

IMPORTANT

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Three lines are given by

$\vec{r} = λ \hat{i}, λ \in R$ ,

$\vec{r} = μ (\hat{i} + \hat{j}), μ \in R$ and

$\vec{r} = ν (\hat{i} + \hat{j} + \hat{k}), ν \in R .$

Let the lines cut the plane $x + y + z = 1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ is $Δ$ then the value of ${(6 Δ)}^{2}$ equals_________

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Important Questions on Three Dimensional Geometry

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2018>Q 10

Let

P_{1} : 2 x + y - z = 3 a n d P_{2} : x + 2 y + z = 2

be two planes. Then, which of the following statement(s) is (are) TRUE?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2016>Q 11

Consider a pyramid

O P Q R 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 ‐ a x i s

and the

y ‐ a x i s

, respectively. The base

O P Q R

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

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2015>Q 12

R^{3}

, consider the planes

P_{1} : y = 0

and

P_{2} : x + z = 1 .

Let

P_{3}

, be a plane, different from

P_{1}

and

P_{2}

, which passes through the intersection of

P_{1}

and

P_{2}

. If the distance of the point

(0, 1, 0)

, from

P_{3}

1

and the distance of a point

(α, β, γ)

, from

P_{3}

2

, then which of the following relations is(are) true ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2015>Q 13

R^{3},

let

L

be a straight line passing through the origin. Suppose that all the points on

L

are at a constant distance from the two planes

P_{1} : x + 2 y - z + 1 = 0

and

P_{2} : 2 x - y + z - 1 = 0 .

Let

M

be the locus of the foot of the perpendiculars drawn from the points on

L

to the plane

P_{1} .

Which of the following point(s) lie(s) on

M ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2014>Q 14

From a point

P (λ, λ, λ),

perpendiculars

P Q

and

P R

are drawn respectively on the lines

y = x, z = 1

and

y = - x, z = - 1

. If

P

is such that

∠ Q P R

is a right angle, then the possible value (s) of

λ

is (are)

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2013>Q 15

Perpendiculars are drawn from points on the line

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

to the plane

x + y + z = 3

. The feet of perpendiculars lie on the line

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Three Dimensional Geometry>JEE Advanced Paper 1 - 2013>Q 16

A line

l

passing through the origin is perpendicular to the lines

l_{1} : (3 + t) \hat{i} + (- 1 + 2 t) \hat{j} + (4 + 2 t) \hat{k}, - \infty < t < \infty

l_{2} : (3 + 2 s) \hat{i} + (3 + 2 s) \hat{j} + (2 + s) \hat{k}, - \infty < s < \infty

Then, the coordinate(s) of the point(s) on

l_{2}

at a distance of

\sqrt{17}

from the point of intersection of

l

and

l_{1}

is (are)

Three lines are given by r→=λi^, λ∈R, r→=μi^+j^,μ∈R and r→=νi^+j^+k^,ν∈R. Let the lines cut the plane x+y+z=1 at the points A, B and C respectively. If the area of the triangle ABC is Δ then the value of 6Δ2 equals_________

Important Questions on Three Dimensional Geometry

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