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3D Geometry

434 Questions
All Exams JEE Mains JEE Advanced
2022 JEE Mains MCQ
JEE Main 2022 (Online) 25th June Morning Shift

Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, $-$1, $-$1), parallel to the line PQ meets the plane S at R, then QR2 is equal to :

A.
2
B.
5
C.
7
D.
11
2022 JEE Mains MCQ
JEE Main 2022 (Online) 24th June Evening Shift

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

A.
16
B.
6
C.
12
D.
15
2022 JEE Mains MCQ
JEE Main 2022 (Online) 24th June Evening Shift

Let the points on the plane P be equidistant from the points ($-$4, 2, 1) and (2, $-$2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is :

A.
${\pi \over 6}$
B.
${\pi \over 4}$
C.
${\pi \over 3}$
D.
${5\pi \over 12}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 1st September Evening Shift
Let the acute angle bisector of the two planes x $-$ 2y $-$ 2z + 1 = 0 and 2x $-$ 3y $-$ 6z + 1 = 0 be the plane P. Then which of the following points lies on P?
A.
$\left( {3,1, - {1 \over 2}} \right)$
B.
$\left( { - 2,0, - {1 \over 2}} \right)$
C.
(0, 2, $-$4)
D.
(4, 0, $-$2)
2021 JEE Mains MCQ
JEE Main 2021 (Online) 1st September Evening Shift
The distance of line $3y - 2z - 1 = 0 = 3x - z + 4$ from the point (2, $-$1, 6) is :
A.
$\sqrt {26} $
B.
$2\sqrt 5 $
C.
$2\sqrt 6 $
D.
$4\sqrt 2 $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 31st August Evening Shift
The distance of the point ($-$1, 2, $-$2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x $-$ 2y + z = 0 is :
A.
${1 \over {\sqrt 2 }}$
B.
${5 \over 2}$
C.
${{\sqrt {42} } \over 2}$
D.
${{\sqrt {34} } \over 2}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 31st August Morning Shift
Let the equation of the plane, that passes through the point (1, 4, $-$3) and contains the line of intersection of the
planes 3x $-$ 2y + 4z $-$ 7 = 0
and x + 5y $-$ 2z + 9 = 0, be
$\alpha$x + $\beta$y + $\gamma$z + 3 = 0, then $\alpha$ + $\beta$ + $\gamma$ is equal to :
A.
$-$23
B.
$-$15
C.
23
D.
15
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th August Evening Shift
The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $-$ n = 0 and mn + nl + lm = 0, is :
A.
${\pi \over 2}$
B.
$\pi - {\cos ^{ - 1}}\left( {{4 \over 9}} \right)$
C.
${\cos ^{ - 1}}\left( {{8 \over 9}} \right)$
D.
${\pi \over 3}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th August Evening Shift
The equation of the plane passing through the line of intersection of the planes $\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1$ and $\overrightarrow r .\left( {2\widehat i + 3\widehat j - \widehat k} \right) + 4 = 0$ and parallel to the x-axis is :
A.
$\overrightarrow r .\left( {\widehat j - 3\widehat k} \right) + 6 = 0$
B.
$\overrightarrow r .\left( {\widehat i + 3\widehat k} \right) + 6 = 0$
C.
$\overrightarrow r .\left( {\widehat i - 3\widehat k} \right) + 6 = 0$
D.
$\overrightarrow r .\left( {\widehat j - 3\widehat k} \right) - 6 = 0$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th August Morning Shift
The distance of the point (1, $-$2, 3) from the plane x $-$ y + z = 5 measured parallel to a line, whose direction ratios are 2, 3, $-$6 is :
A.
3
B.
5
C.
2
D.
1
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th August Morning Shift
Equation of a plane at a distance $\sqrt {{2 \over {21}}} $ from the origin, which contains the line of intersection of the planes x $-$ y $-$ z $-$ 1 = 0 and 2x + y $-$ 3z + 4 = 0, is :
A.
$3x - y - 5z + 2 = 0$
B.
$3x - 4z + 3 = 0$
C.
$ - x + 2y + 2z - 3 = 0$
D.
$4x - y - 5z + 2 = 0$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th August Evening Shift
Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes $\overrightarrow r \,.\,\left( {\widehat i + \widehat j + 4\widehat k} \right) = 16$ and $\overrightarrow r \,.\,\left( { - \widehat i + \widehat j + \widehat k} \right) = 6$. Then which of the following points does NOT lie on P?
A.
(3, 3, 2)
B.
(6, $-$6, 2)
C.
(4, 2, 2)
D.
($-$8, 8, 6)
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th August Morning Shift
A plane P contains the line $x + 2y + 3z + 1 = 0 = x - y - z - 6$, and is perpendicular to the plane $ - 2x + y + z + 8 = 0$. Then which of the following points lies on P?
A.
($-$1, 1, 2)
B.
(0, 1, 1)
C.
(1, 0, 1)
D.
(2, $-$1, 1)
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th July Evening Shift
For real numbers $\alpha$ and $\beta$ $\ne$ 0, if the point of intersection of the straight lines

${{x - \alpha } \over 1} = {{y - 1} \over 2} = {{z - 1} \over 3}$ and ${{x - 4} \over \beta } = {{y - 6} \over 3} = {{z - 7} \over 3}$, lies on the plane x + 2y $-$ z = 8, then $\alpha$ $-$ $\beta$ is equal to :
A.
5
B.
9
C.
3
D.
7
2021 JEE Mains MCQ
JEE Main 2021 (Online) 27th July Morning Shift
Let the plane passing through the point ($-$1, 0, $-$2) and perpendicular to each of the planes 2x + y $-$ z = 2 and x $-$ y $-$ z = 3 be ax + by + cz + 8 = 0. Then the value of a + b + c is equal to :
A.
3
B.
8
C.
5
D.
4
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th July Morning Shift
Let the foot of perpendicular from a point P(1, 2, $-$1) to the straight line $L:{x \over 1} = {y \over 0} = {z \over { - 1}}$ be N. Let a line be drawn from P parallel to the plane x + y + 2z = 0 which meets L at point Q. If $\alpha$ is the acute angle between the lines PN and PQ, then cos$\alpha$ is equal to ________________.
A.
${1 \over {\sqrt 5 }}$
B.
${{\sqrt 3 } \over 2}$
C.
${1 \over {\sqrt 3 }}$
D.
${1 \over {2\sqrt 3 }}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 22th July Evening Shift
Let L be the line of intersection of planes $\overrightarrow r .(\widehat i - \widehat j + 2\widehat k) = 2$ and $\overrightarrow r .(2\widehat i + \widehat j - \widehat k) = 2$. If $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular on L from the point (1, 2, 0), then the value of $35(\alpha + \beta + \gamma )$ is equal to :
A.
101
B.
119
C.
143
D.
134
2021 JEE Mains MCQ
JEE Main 2021 (Online) 22th July Evening Shift
If the shortest distance between the straight lines $3(x - 1) = 6(y - 2) = 2(z - 1)$ and $4(x - 2) = 2(y - \lambda ) = (z - 3),\lambda \in R$ is ${1 \over {\sqrt {38} }}$, then the integral value of $\lambda$ is equal to :
A.
3
B.
2
C.
5
D.
$-$1
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Evening Shift
The lines x = ay $-$ 1 = z $-$ 2 and x = 3y $-$ 2 = bz $-$ 2, (ab $\ne$ 0) are coplanar, if :
A.
b = 1, a$\in$R $-$ {0}
B.
a = 1, b$\in$R $-$ {0}
C.
a = 2, b = 2
D.
a = 2, b = 3
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Evening Shift
Consider the line L given by the equation

${{x - 3} \over 2} = {{y - 1} \over 1} = {{z - 2} \over 1}$.

Let Q be the mirror image of the point (2, 3, $-$1) with respect to L. Let a plane P be such that it passes through Q, and the line L is perpendicular to P. Then which of the following points is on the plane P?
A.
($-$1, 1, 2)
B.
(1, 1, 1)
C.
(1, 1, 2)
D.
(1, 2, 2)
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Evening Shift
If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line ${{x + 1} \over 2} = {{y - 3} \over 1} = {{z + 2} \over { - 1}}$ and containing the line ${{x - 2} \over 3} = {{1 - y} \over 2} = {{z + 1} \over 1}$ is $\alpha$x + $\beta$y + $\gamma$z = 24, then $\alpha$ + $\beta$ + $\gamma$ is equal to :
A.
21
B.
19
C.
18
D.
20
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Morning Shift
The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :
A.
x + 3z = 0
B.
3x $-$ z = 0
C.
x + 3z = 10
D.
3x + z = 6
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Evening Shift
If the foot of the perpendicular from point (4, 3, 8) on the line ${L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}$, l $\ne$ 0 is (3, 5, 7), then the shortest distance between the line L1 and line ${L_2}:{{x - 2} \over 3} = {{y - 4} \over 4} = {{z - 5} \over 5}$ is equal to :
A.
${1 \over {\sqrt 6 }}$
B.
${1 \over 2}$
C.
${1 \over {\sqrt 3 }}$
D.
$\sqrt {{2 \over 3}} $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Evening Shift
If (x, y, z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of the expression
$3 + {{x - 11} \over {{{(y - 19)}^2}{{(z - 12)}^2}}} + {{y - 19} \over {{{(x - 11)}^2}{{(z - 12)}^2}}} + {{z - 12} \over {{{(x - 11)}^2}{{(y - 19)}^2}}} - {{x + y + z} \over {14(x - 11)(y - 19)(z - 12)}}$ is equal to :
A.
3
B.
39
C.
$-$45
D.
0
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Morning Shift
Let the position vectors of two points P and Q be 3$\widehat i$ $-$ $\widehat j$ + 2$\widehat k$ and $\widehat i$ + 2$\widehat j$ $-$ 4$\widehat k$, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, $-$1, 2) and ($-$2, 1, $-$2), respectively. Let lines PR and QS intersect at T. If the vector $\overrightarrow {TA} $ is perpendicular to both $\overrightarrow {PR} $ and $\overrightarrow {QS} $ and the length of vector $\overrightarrow {TA} $ is $\sqrt 5 $ units, then the modulus of a position vector of A is :
A.
$\sqrt {171} $
B.
$\sqrt {227} $
C.
$\sqrt {482} $
D.
$\sqrt {5} $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Morning Shift
Let P be a plane lx + my + nz = 0 containing

the line, ${{1 - x} \over 1} = {{y + 4} \over 2} = {{z + 2} \over 3}$. If plane P divides the line segment AB joining

points A($-$3, $-$6, 1) and B(2, 4, $-$3) in ratio k : 1 then the value of k is equal to :
A.
2
B.
3
C.
1.5
D.
4
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Morning Shift
If for a > 0, the feet of perpendiculars from the points A(a, $-$2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, $-$a, $-$1) and D respectively, then the length of line segment CD is equal to :
A.
$\sqrt {41} $
B.
$\sqrt {55} $
C.
$\sqrt {31} $
D.
$\sqrt {66} $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
If the mirror image of the point (1, 3, 5) with respect to the plane

4x $-$ 5y + 2z = 8 is ($\alpha$, $\beta$, $\gamma$), then 5($\alpha$ + $\beta$ + $\gamma$) equals :
A.
39
B.
41
C.
47
D.
43
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4. If point P($\alpha$, $\beta$, $\gamma$) is the foot of perpendicular from (3, 2, 1) on L, then the
value of 21($\alpha$ + $\beta$ + $\gamma$) equals :
A.
102
B.
142
C.
136
D.
68
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Morning Shift
Consider the three planes

P1 : 3x + 15y + 21z = 9,

P2 : x $-$ 3y $-$ z = 5, and

P3 : 2x + 10y + 14z = 5

Then, which one of the following is true?
A.
P1 and P2 are parallel.
B.
P1, P2 and P3 all are parallel.
C.
P1 and P3 are parallel.
D.
P2 and P3 are parallel.
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Morning Shift
If (1, 5, 35), (7, 5, 5), (1, $\lambda$, 7) and (2$\lambda$, 1, 2) are coplanar, then the sum of all possible values of $\lambda$ is :
A.
$ - {{44} \over 5}$
B.
$ - {{39} \over 5}$
C.
${{44} \over 5}$
D.
${{39} \over 5}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Evening Shift
A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, $-$1, 1), then the projection of $\overrightarrow {OP} $ on this plane is of length :
A.
$\sqrt {{2 \over 7}} $
B.
$\sqrt {{2 \over 5}} $
C.
$\sqrt {{2 \over 3}} $
D.
$\sqrt {{2 \over 11}} $
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Morning Shift
The equation of the line through the point (0, 1, 2) and perpendicular to the line

${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over { - 2}}$ is :
A.
${x \over 3} = {{y - 1} \over { - 4}} = {{z - 2} \over 3}$
B.
${x \over 3} = {{y - 1} \over 4} = {{z - 2} \over { - 3}}$
C.
${x \over { - 3}} = {{y - 1} \over 4} = {{z - 2} \over 3}$
D.
${x \over 3} = {{y - 1} \over 4} = {{z - 2} \over 3}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Morning Shift
Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations l + m $-$ n = 0 and l2 + m2 $-$ n2 = 0. Then the value of sin4$\alpha$ + cos4$\alpha$ is :
A.
${{3 \over 8}}$
B.
${{3 \over 4}}$
C.
${{1 \over 2}}$
D.
${{5 \over 8}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 24th February Evening Shift
Let a, b$ \in $R. If the mirror image of the point P(a, 6, 9) with respect to the line

${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$ is (20, b, $-$a$-$9), then | a + b |, is equal to :
A.
88
B.
90
C.
86
D.
84
2021 JEE Mains MCQ
JEE Main 2021 (Online) 24th February Evening Shift
The vector equation of the plane passing through the intersection

of the planes $\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1$ and $\overrightarrow r .\left( {\widehat i - 2\widehat j} \right) = - 2$, and the point (1, 0, 2) is :
A.
$\overrightarrow r .\left( {\widehat i + 7\widehat j + 3\widehat k} \right) = {7 \over 3}$
B.
$\overrightarrow r .\left( {\widehat i + 7\widehat j + 3\widehat k} \right) = 7$
C.
$\overrightarrow r .\left( {3\widehat i + 7\widehat j + 3\widehat k} \right) = 7$
D.
$\overrightarrow r .\left( {\widehat i - 7\widehat j + 3\widehat k} \right) = {7 \over 3}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 24th February Morning Shift
The equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes

3x + y - 2z = 5 and 2x - 5y - z = 7, is :
A.
6x - 5y + 2z + 10 =0
B.
3x - 10y - 2z + 11 = 0
C.
6x - 5y - 2z - 2 = 0
D.
11x + y + 17z + 38 = 0
2021 JEE Mains MCQ
JEE Main 2021 (Online) 24th February Morning Shift
The distance of the point (1, 1, 9) from the point of intersection of the line ${{x - 3} \over 1} = {{y - 4} \over 2} = {{z - 5} \over 2}$ and the plane x + y + z = 17 is :
A.
$19\sqrt 2 $
B.
$2\sqrt {19} $
C.
38
D.
$\sqrt {38} $
2020 JEE Mains MCQ
JEE Main 2020 (Online) 6th September Evening Slot
A plane P meets the coordinate axes at A, B and C respectively. The centroid of $\Delta $ABC is given to be (1, 1, 2). Then the equation of the line through this centroid and perpendicular to the plane P is :
A.
${{x - 1} \over 1} = {{y - 1} \over 1} = {{z - 2} \over 2}$
B.
${{x - 1} \over 2} = {{y - 1} \over 1} = {{z - 2} \over 1}$
C.
${{x - 1} \over 2} = {{y - 1} \over 2} = {{z - 2} \over 1}$
D.
${{x - 1} \over 1} = {{y - 1} \over 2} = {{z - 2} \over 2}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 6th September Morning Slot
The shortest distance between the lines

${{x - 1} \over 0} = {{y + 1} \over { - 1}} = {z \over 1}$

and x + y + z + 1 = 0, 2x – y + z + 3 = 0 is :
A.
1
B.
${1 \over 2}$
C.
${1 \over {\sqrt 2 }}$
D.
${1 \over {\sqrt 3 }}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Evening Slot
If for some $\alpha $ $ \in $ R, the lines

L1 : ${{x + 1} \over 2} = {{y - 2} \over { - 1}} = {{z - 1} \over 1}$ and

L2 : ${{x + 2} \over \alpha } = {{y + 1} \over {5 - \alpha }} = {{z + 1} \over 1}$ are coplanar,

then the line L2 passes through the point :
A.
(10, 2, 2)
B.
(2, –10, –2)
C.
(10, –2, –2)
D.
(–2, 10, 2)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Morning Slot
If (a, b, c) is the image of the point (1, 2, -3) in

the line ${{x + 1} \over 2} = {{y - 3} \over { - 2}} = {z \over { - 1}}$, then a + b + c is :
A.
1
B.
2
C.
3
D.
-1
2020 JEE Mains MCQ
JEE Main 2020 (Online) 4th September Evening Slot
The distance of the point (1, –2, 3) from

the plane x – y + z = 5 measured parallel to

the line ${x \over 2} = {y \over 3} = {z \over { - 6}}$ is :
A.
7
B.
1
C.
${1 \over 7}$
D.
${7 \over 5}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 3rd September Evening Slot
The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right angles also passes through the point :
A.
(4, 0, 1)
B.
(0, –1, 1)
C.
(0, 1, –1)
D.
(4, 0, –1)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 3rd September Morning Slot
The foot of the perpendicular drawn from the point (4, 2, 3) to the line joining the points (1, –2, 3) and (1, 1, 0) lies on the plane :
A.
x – 2y + z = 1
B.
x + 2y – z = 1
C.
x – y – 2y = 1
D.
2x + y – z = 1
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
A plane passing through the point (3, 1, 1) contains two lines whose direction ratios are 1, –2, 2 and 2, 3, –1 respectively. If this plane also passes through the point ($\alpha $, –3, 5), then $\alpha $ is equal to:
A.
-10
B.
10
C.
5
D.
-5
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Morning Slot
The plane passing through the points (1, 2, 1),
(2, 1, 2) and parallel to the line, 2x = 3y, z = 1
also passes through the point :
A.
(0, 6, –2)
B.
(–2, 0, 1)
C.
(0, –6, 2)
D.
(2, 0 –1)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Evening Slot
The mirror image of the point (1, 2, 3) in a plane is

$\left( { - {7 \over 3}, - {4 \over 3}, - {1 \over 3}} \right)$. Which of the following points lies on this plane ?
A.
(1, –1, 1)
B.
(–1, –1, –1)
C.
(–1, –1, 1)
D.
(1, 1, 1)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Morning Slot
The shortest distance between the lines

${{x - 3} \over 3} = {{y - 8} \over { - 1}} = {{z - 3} \over 1}$ and

${{x + 3} \over { - 3}} = {{y + 7} \over 2} = {{z - 6} \over 4}$ is :
A.
3
B.
${7 \over 2}\sqrt {30} $
C.
$3\sqrt {30} $
D.
$2\sqrt {30} $
2020 JEE Mains MCQ
JEE Main 2020 (Online) 7th January Morning Slot
Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is :
A.
(4, 3, 2)
B.
(6, 5, - 2)
C.
(3, 4, -2)
D.
(6, 5, 2)