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3D Geometry

390 Questions Start JEE Mains Test
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2019 Q301 JEE Mains MCQ
14 Mar 2026
A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0, passes through the point :
A.
(1, –4, 1)
B.
(1, 4, –1)
C.
(2, 4, 1)
D.
(2, –4, 1)
2019 Q302 JEE Mains MCQ
14 Mar 2026
If the line ${{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 1} \over { - 1}}$ intersects the plane 2x + 3y – z + 13 = 0 at a point P and the plane 3x + y + 4z = 16 at a point Q, then PQ is equal to :
A.
$2\sqrt 7 $
B.
14
C.
$2\sqrt {14} $
D.
$\sqrt {14} $
2019 Q303 JEE Mains MCQ
14 Mar 2026
A perpendicular is drawn from a point on the line ${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {z \over 1}$ to the plane x + y + z = 3 such that the foot of the perpendicular Q also lies on the plane x – y + z = 3. Then the co-ordinates of Q are :
A.
(4, 0, – 1)
B.
(2, 0, 1)
C.
(1, 0, 2)
D.
(– 1, 0, 4)
2019 Q304 JEE Mains MCQ
14 Mar 2026
If the plane 2x – y + 2z + 3 = 0 has the distances ${1 \over 3}$ and ${2 \over 3}$ units from the planes 4x – 2y + 4z + $\lambda $ = 0 and 2x – y + 2z + $\mu $ = 0, respectively, then the maximum value of $\lambda $ + $\mu $ is equal to :
A.
13
B.
9
C.
5
D.
15
2019 Q305 JEE Mains MCQ
14 Mar 2026
If the length of the perpendicular from the point ($\beta $, 0, $\beta $) ($\beta $ $ \ne $ 0) to the line,
${x \over 1} = {{y - 1} \over 0} = {{z + 1} \over { - 1}}$ is $\sqrt {{3 \over 2}} $, then $\beta $ is equal to :
A.
2
B.
1
C.
-2
D.
-1
2019 Q306 JEE Mains MCQ
14 Mar 2026
If Q(0, –1, –3) is the image of the point P in the plane 3x – y + 4z = 2 and R is the point (3, –1, –2), then the area (in sq. units) of $\Delta $PQR is :
A.
${{\sqrt {65} } \over 2}$
B.
$2\sqrt {13} $
C.
${{\sqrt {91} } \over 2}$
D.
${{\sqrt {91} } \over 4}$
2019 Q307 JEE Mains MCQ
14 Mar 2026
The vertices B and C of a $\Delta $ABC lie on the line,

${{x + 2} \over 3} = {{y - 1} \over 0} = {z \over 4}$ such that BC = 5 units.

Then the area (in sq. units) of this triangle, given that the point A(1, –1, 2), is :
A.
6
B.
$5\sqrt {17} $
C.
$\sqrt {34} $
D.
$2\sqrt {34} $
2019 Q308 JEE Mains MCQ
14 Mar 2026
Let P be the plane, which contains the line of intersection of the planes, x + y + z – 6 = 0 and 2x + 3y + z + 5 = 0 and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to :
A.
205$\sqrt5$
B.
63$\sqrt5$
C.
11/$\sqrt5$
D.
17/$\sqrt5$
2019 Q309 JEE Mains MCQ
14 Mar 2026
A plane passing through the points (0, –1, 0) and (0, 0, 1) and making an angle ${\pi \over 4}$ with the plane y – z + 5 = 0, also passes through the point
A.
$\left( {\sqrt 2 ,1,4} \right)$
B.
$\left(- {\sqrt 2 ,1,4} \right)$
C.
$\left( -{\sqrt 2 ,-1,-4} \right)$
D.
$\left( {\sqrt 2 ,-1,4} \right)$
2019 Q310 JEE Mains MCQ
14 Mar 2026
If the line, ${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 2} \over 4}$ meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is :
A.
${{\sqrt 5 } \over 2}$
B.
2$\sqrt 5$
C.
9/2
D.
7/2
2019 Q311 JEE Mains MCQ
14 Mar 2026
If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the distance of R from the origin is :
A.
$2 \sqrt {14}$
B.
$ \sqrt {53}$
C.
$2 \sqrt {21}$
D.
6
2019 Q312 JEE Mains MCQ
14 Mar 2026
The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y+ 4z = 5 which is perpendicular to the plane x – y + z = 0 is :
A.
$\mathop r\limits^ \to \times \left( {\mathop i\limits^ \wedge - \mathop k\limits^ \wedge } \right) - 2 = 0$
B.
$\mathop r\limits^ \to . \left( {\mathop i\limits^ \wedge + \mathop k\limits^ \wedge } \right) + 2 = 0$
C.
$\mathop r\limits^ \to . \left( {\mathop i\limits^ \wedge - \mathop k\limits^ \wedge } \right) + 2 = 0$
D.
$\mathop r\limits^ \to \times \left( {\mathop i\limits^ \wedge - \mathop k\limits^ \wedge } \right) + 2 = 0$
2019 Q313 JEE Mains MCQ
14 Mar 2026
The magnitude of the projection of the vector $\mathop {2i}\limits^ \wedge + \mathop {3j}\limits^ \wedge + \mathop k\limits^ \wedge $ on the vector perpendicular to the plane containing the vectors $\mathop {i}\limits^ \wedge + \mathop {j}\limits^ \wedge + \mathop k\limits^ \wedge $ and $\mathop {i}\limits^ \wedge + \mathop {2j}\limits^ \wedge + \mathop {3k}\limits^ \wedge $ , is :
A.
${{\sqrt 3 } \over 2}$
B.
$\sqrt 6 $
C.
$\sqrt {3 \over 2} $
D.
3$\sqrt 6 $
2019 Q314 JEE Mains MCQ
14 Mar 2026
The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing through the point (1, 1, 0) is :
A.
x – 3y – 2z = –2
B.
2x – z = 2
C.
x – y – z = 0
D.
x + 3y + z = 4
2019 Q315 JEE Mains MCQ
14 Mar 2026
The length of the perpendicular from the point (2, –1, 4) on the straight line,

${{x + 3} \over {10}}$= ${{y - 2} \over {-7}}$ = ${{z} \over {1}}$ is :
A.
less than 2
B.
greater than 4
C.
greater than 2 but less than 3
D.
greater than 3 but less than 4
2019 Q316 JEE Mains MCQ
14 Mar 2026
Let S be the set of all real values of $\lambda $ such that a plane passing through the points (–$\lambda $2, 1, 1), (1, –$\lambda $2, 1) and (1, 1, – $\lambda $2) also passes through the point (–1, –1, 1). Then S is equal to :
A.
{1, $-$1}
B.
{3, $-$ 3}
C.
$\left\{ {\sqrt 3 } \right\}$
D.
$\left\{ {\sqrt 3 , - \sqrt 3 } \right\}$
2019 Q317 JEE Mains MCQ
14 Mar 2026
If an angle between the line, ${{x + 1} \over 2} = {{y - 2} \over 1} = {{z - 3} \over { - 2}}$ and the plane, $x - 2y - kz = 3$ is ${\cos ^{ - 1}}\left( {{{2\sqrt 2 } \over 3}} \right),$ then a value of k is :
A.
$\sqrt {{3 \over 5}} $
B.
$ - {5 \over 2}$
C.
$ - {3 \over 2}$
D.
$\sqrt {{5 \over 3}} $
2019 Q318 JEE Mains MCQ
14 Mar 2026
The perpendicular distance from the origin to the plane containing the two lines,

${{x + 2} \over 3} = {{y - 2} \over 5} = {{z + 5} \over 7}$ and

${{x - 1} \over 1} = {{y - 4} \over 4} = {{z + 4} \over 7},$ is :
A.
$6\sqrt {11} $
B.
${{11} \over {\sqrt 6 }}$
C.
11
D.
11$\sqrt 6 $
2019 Q319 JEE Mains MCQ
14 Mar 2026
A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(–1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is :
A.
cos$-$1$\left( {{{17} \over {31}}} \right)$
B.
cos$-$1$\left( {{{9} \over {35}}} \right)$
C.
cos$-$1$\left( {{{19} \over {35}}} \right)$
D.
cos$-$1$\left( {{7 \over {31}}} \right)$
2019 Q320 JEE Mains MCQ
14 Mar 2026
Two lines ${{x - 3} \over 1} = {{y + 1} \over 3} = {{z - 6} \over { - 1}}$ and ${{x + 5} \over 7} = {{y - 2} \over { - 6}} = {{z - 3} \over 4}$ intersect at the point R. The reflection of R in the xy-plane has coordinates :
A.
(2, 4, 7)
B.
(2, $-$ 4, $-$7)
C.
(2, $-$ 4, 7)
D.
($-$ 2, 4, 7)
2019 Q321 JEE Mains MCQ
14 Mar 2026
If the point (2, $\alpha $, $\beta $) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane 2x – 5y = 15, then 2$\alpha $ – 3$\beta $ is equal to
A.
12
B.
7
C.
17
D.
5
2019 Q322 JEE Mains MCQ
14 Mar 2026
The plane containing the line ${{x - 3} \over 2} = {{y + 2} \over { - 1}} = {{z - 1} \over 3}$ and also containing its projection on the plane 2x + 3y $-$ z = 5, contains which one of the following points ?
A.
($-$ 2, 2, 2)
B.
(2, 2, 0)
C.
(2, 0, $-$ 2)
D.
(0, $-$ 2, 2)
2019 Q323 JEE Mains MCQ
14 Mar 2026
The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an angle ${\pi \over 4}$ with the plane y $-$ z + 5 = 0 are :
A.
2, $-$1, 1
B.
$2\sqrt 3 ,1, - 1$
C.
$\sqrt 2 ,1, - 1$
D.
$\sqrt 2 , - \sqrt 2 $
2019 Q324 JEE Mains MCQ
14 Mar 2026
On which of the following lines lies the point of intersection of the line,   ${{x - 4} \over 2} = {{y - 5} \over 2} = {{z - 3} \over 1}$  and the plane, x + y + z = 2 ?
A.
${{x - 4} \over 1} = {{y - 5} \over 1} = {{z - 5} \over { - 1}}$
B.
${{x - 2} \over 2} = {{y - 3} \over 2} = {{z + 3} \over 3}$
C.
${{x - 1} \over 1} = {{y - 3} \over 2} = {{z + 4} \over { - 5}}$
D.
${{x + 3} \over 3} = {{4 - y} \over 3} = {{z + 1} \over { - 2}}$
2019 Q325 JEE Mains MCQ
14 Mar 2026
The plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles, passes through which one of the following points ?
A.
(2, 1, 3)
B.
(4, $-$ 1, 2)
C.
(4, 1, $-$ 2)
D.
($-$ 2, 3, 5)
2019 Q326 JEE Mains MCQ
14 Mar 2026
The plane passing through the point (4, –1, 2) and parallel to the lines  ${{x + 2} \over 3} = {{y - 2} \over { - 1}} = {{z + 1} \over 2}$  and  ${{x - 2} \over 1} = {{y - 3} \over 2} = {{z - 4} \over 3}$ also passes through the point -
A.
(1, 1, $-$ 1)
B.
(1, 1, 1)
C.
($-$ 1, $-$ 1, $-$1)
D.
($-$ 1, $-$ 1, 1)
2019 Q327 JEE Mains MCQ
14 Mar 2026
Let A be a point on the line $\overrightarrow r = \left( {1 - 3\mu } \right)\widehat i + \left( {\mu - 1} \right)\widehat j + \left( {2 + 5\mu } \right)\widehat k$ and B(3, 2, 6) be a point in the space. Then the value of $\mu $ for which the vector $\overrightarrow {AB} $  is parallel to the plane x $-$ 4y + 3z = 1 is -
A.
${1 \over 8}$
B.
${1 \over 2}$
C.
${1 \over 4}$
D.
$-$ ${1 \over 4}$
2019 Q328 JEE Mains MCQ
14 Mar 2026
The equation of the plane containing the straight line ${x \over 2} = {y \over 3} = {z \over 4}$ and perpendicular to the plane containing the straight lines ${x \over 3} = {y \over 4} = {z \over 2}$ and ${x \over 4} = {y \over 2} = {z \over 3}$ is :
A.
x $-$ 2y + z = 0
B.
3x + 2y $-$ 3z = 0
C.
x + 2y $-$ 2z = 0
D.
5x + 2y $-$ 4z = 0
2019 Q329 JEE Mains MCQ
14 Mar 2026
If the lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then :
A.
ab'  +  bc'  +  1  =  0
B.
cc'  +  a   +  a'  =  0
C.
bb'  +  cc'  +  1  =  0
D.
aa'  +  c  +  c'  =  0
2019 Q330 JEE Mains MCQ
14 Mar 2026
The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to y-axis also passes through the point :
A.
(–3, 0, -1)
B.
(3, 2, 1)
C.
(3, 3, -1)
D.
(–3, 1, 1)
2019 Q331 JEE Mains MCQ
14 Mar 2026
The equation of the line passing through (–4, 3, 1), parallel

to the plane x + 2y – z – 5 = 0 and intersecting

the line ${{x + 1} \over { - 3}} = {{y - 3} \over 2} = {{z - 2} \over { - 1}}$ is :
A.
${{x + 4} \over 3} = {{y - 3} \over {-1}} = {{z - 1} \over 1}$
B.
${{x + 4} \over 1} = {{y - 3} \over {1}} = {{z - 1} \over 3}$
C.
${{x + 4} \over -1} = {{y - 3} \over {1}} = {{z - 1} \over 1}$
D.
${{x - 4} \over 2} = {{y + 3} \over {1}} = {{z + 1} \over 4}$
2018 Q332 JEE Mains MCQ
14 Mar 2026
The sum of the intercepts on the coordinate axes of the plane passing through the point ($-$2, $-2,$ 2) and containing the line joining the points (1, $-$1, 2) and (1, 1, 1) is :
A.
4
B.
$-$ 4
C.
$-$ 8
D.
12
2018 Q333 JEE Mains MCQ
14 Mar 2026
If the angle between the lines, ${x \over 2} = {y \over 2} = {z \over 1}$

and ${{5 - x} \over { - 2}} = {{7y - 14} \over p} = {{z - 3} \over 4}\,\,$ is ${\cos ^{ - 1}}\left( {{2 \over 3}} \right),$ then p is equal to :
A.
${7 \over 2}$
B.
${2 \over 7}$
C.
$-$ ${7 \over 4}$
D.
$-$ ${4 \over 7}$
2018 Q334 JEE Mains MCQ
14 Mar 2026
The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x + y + z = 7 is :
A.
$\sqrt {{2 \over 3}} $
B.
${2 \over {\sqrt 3 }}$
C.
${2 \over 3}$
D.
${1 \over 3}$
2018 Q335 JEE Mains MCQ
14 Mar 2026
If L1 is the line of intersection of the planes 2x - 2y + 3z - 2 = 0, x - y + z + 1 = 0 and L2 is the line of intersection of the planes x + 2y - z - 3 = 0, 3x - y + 2z - 1 = 0, then the distance of the origin from the plane, containing the lines L1 and L2, is :
A.
${1 \over {\sqrt 2 }}$
B.
${1 \over {4\sqrt 2 }}$
C.
${1 \over {3\sqrt 2 }}$
D.
${1 \over {2\sqrt 2 }}$
2018 Q336 JEE Mains MCQ
14 Mar 2026
An angle between the lines whose direction cosines are gien by the equations,
$l$ + 3m + 5n = 0 and 5$l$m $-$ 2mn + 6n$l$ = 0, is :
A.
${\cos ^{ - 1}}\left( {{1 \over 3}} \right)$
B.
${\cos ^{ - 1}}\left( {{1 \over 4}} \right)$
C.
${\cos ^{ - 1}}\left( {{1 \over 6}} \right)$
D.
${\cos ^{ - 1}}\left( {{1 \over 8}} \right)$
2018 Q337 JEE Mains MCQ
14 Mar 2026
A plane bisects the line segment joining the points (1, 2, 3) and ($-$ 3, 4, 5) at rigt angles. Then this plane also passes through the point :
A.
($-$ 3, 2, 1)
B.
(3, 2, 1)
C.
($-$ 1, 2, 3)
D.
(1, 2, $-$ 3)
2018 Q338 JEE Mains MCQ
14 Mar 2026
A variable plane passes through a fixed point (3,2,1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz -plane through A, a second plane is drawn parallel zx-plane through B and a third plane is drawn parallel to xy-plane through C. Then the locus of the point of intersection of these three planes, is :
A.
${x \over 3} + {y \over 2} + {z \over 1} = 1$
B.
x + y + z = 6
C.
${1 \over x} + {1 \over y} + {1 \over z} = {{11} \over 6}$
D.
${3 \over x} + {2 \over y} + {1 \over z} = 1$
2018 Q339 JEE Mains MCQ
14 Mar 2026
An angle between the plane, x + y + z = 5 and the line of intersection of the planes, 3x + 4y + z $-$ 1 = 0 and 5x + 8y + 2z + 14 =0, is :
A.
${\sin ^{ - 1}}\left( {\sqrt {{\raise0.5ex\hbox{$\scriptstyle 3$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {17}$}}} } \right)$
B.
${\cos ^{ - 1}}\left( {\sqrt {{\raise0.5ex\hbox{$\scriptstyle 3$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {17}$}}} } \right)$
C.
${\cos ^{ - 1}}\left( {{\raise0.5ex\hbox{$\scriptstyle 3$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {17}$}}} \right)$
D.
${\sin ^{ - 1}}\left( {{\raise0.5ex\hbox{$\scriptstyle 3$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {17}$}}} \right)$
2017 Q340 JEE Mains MCQ
14 Mar 2026
If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of $\Delta $ABC is :
A.
${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = 1$
B.
${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = 3$
C.
${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = {1 \over 9}$
D.
${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = 9$
2017 Q341 JEE Mains MCQ
14 Mar 2026
If the line, ${{x - 3} \over 1} = {{y + 2} \over { - 1}} = {{z + \lambda } \over { - 2}}$ lies in the plane, 2x−4y+3z=2, then the shortest distance between this line and the line, ${{x - 1} \over {12}} = {y \over 9} = {z \over 4}$ is :
A.
2
B.
1
C.
0
D.
3
2017 Q342 JEE Mains MCQ
14 Mar 2026
If x = a, y = b, z = c is a solution of the system of linear equations

x + 8y + 7z = 0

9x + 2y + 3z = 0

x + y + z = 0

such that the point (a, b, c) lies on the plane x + 2y + z = 6, then 2a + b + c equals :
A.
$-$ 1
B.
0
C.
1
D.
2
2017 Q343 JEE Mains MCQ
14 Mar 2026
The line of intersection of the planes $\overrightarrow r .\left( {3\widehat i - \widehat j + \widehat k} \right) = 1\,\,$ and
$\overrightarrow r .\left( {\widehat i + 4\widehat j - 2\widehat k} \right) = 2,$ is :
A.
${{x - {4 \over 7}} \over { - 2}} = {y \over 7} = {{z - {5 \over 7}} \over {13}}$
B.
${{x - {4 \over 7}} \over 2} = {y \over { - 7}} = {{z + {5 \over 7}} \over {13}}$
C.
${{x - {6 \over {13}}} \over 2} = {{y - {5 \over {13}}} \over { - 7}} = {z \over { - 13}}$
D.
${{x - {6 \over {13}}} \over 2} = {{y - {5 \over {13}}} \over 7} = {z \over { - 13}}$
2017 Q344 JEE Mains MCQ
14 Mar 2026
The coordinates of the foot of the perpendicular from the point (1, $-$2, 1) on the plane containing the lines, ${{x + 1} \over 6} = {{y - 1} \over 7} = {{z - 3} \over 8}$ and ${{x - 1} \over 3} = {{y - 2} \over 5} = {{z - 3} \over 7},$ is :
A.
(2, $-$4, 2)
B.
($-$ 1, 2, $-$1)
C.
(0, 0, 0)
D.
(1, 1, 1)
2017 Q345 JEE Mains MCQ
14 Mar 2026
The distance of the point (1, 3, – 7) from the plane passing through the point (1, –1, – 1), having normal perpendicular to both the lines

${{x - 1} \over 1} = {{y + 2} \over { - 2}} = {{z - 4} \over 3}$

and

${{x - 2} \over 2} = {{y + 1} \over { - 1}} = {{z + 7} \over { - 1}}$ is :
A.
${{10} \over {\sqrt {83} }}$
B.
${{5} \over {\sqrt {83} }}$
C.
${{10} \over {\sqrt {74} }}$
D.
${{20} \over {\sqrt {74} }}$
2017 Q346 JEE Mains MCQ
14 Mar 2026
If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line,

${x \over 1} = {y \over 4} = {z \over 5}$ is Q, then PQ is equal to:
A.
$2\sqrt {42} $
B.
$\sqrt {42} $
C.
$6\sqrt 5 $
D.
$3\sqrt 5 $
2016 Q347 JEE Mains MCQ
14 Mar 2026
The number of distinct real values of $\lambda $ for which the lines

${{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over {{\lambda ^2}}}$ and ${{x - 3} \over 1} = {{y - 2} \over {{\lambda ^2}}} = {{z - 1} \over 2}$ are coplanar is :
A.
4
B.
1
C.
2
D.
3
2016 Q348 JEE Mains MCQ
14 Mar 2026
ABC is a triangle in a plane with vertices

A(2, 3, 5), B(−1, 3, 2) and C($\lambda $, 5, $\mu $).

If the median through A is equally inclined to the coordinate axes, then the value of ($\lambda $3 + $\mu $3 + 5) is :
A.
1130
B.
1348
C.
676
D.
1077
2016 Q349 JEE Mains MCQ
14 Mar 2026
The shortest distance between the lines ${x \over 2} = {y \over 2} = {z \over 1}$ and
${{x + 2} \over { - 1}} = {{y - 4} \over 8} = {{z - 5} \over 4}$ lies in the interval :
A.
[0, 1)
B.
[1, 2)
C.
(2,  3]
D.
(3, 4]
2016 Q350 JEE Mains MCQ
14 Mar 2026
The distance of the point (1, − 2, 4) from the plane passing through the point (1, 2, 2) and perpendicular to the planes x − y + 2z = 3 and 2x − 2y + z + 12 = 0, is :
A.
$2\sqrt 2 $
B.
2
C.
$\sqrt 2 $
D.
${1 \over {\sqrt 2 }}$