The point (0, 3) is nearest to the curve ${x^2} = 2y$ at

$\left( {2\sqrt 2 ,\;0} \right)$

The equation of the locus of a point equidistant from the points $\left( {{a_1},\;{b_1}} \right)$ and $\left( {{a_2},\;{b_2}} \right)$ is $\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0$, then the value of $c$ is

$\sqrt {\left( {a_1^2 + b_1^2 + c_1^2} \right)} $

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JEE • MATHEMATICS • 3D GEOMETRY SYSTEM

3d geometry system practice · Medium

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 53 questions

1 Medium

1 of 53

Application +4 | -1

The equation of the plane perpendicular to the

y z

-plane and passing through the points

(1,-2,4)

and

(3,-4

5)

View answer and explanation

Correct option: C

y+2 z=6

Option D (

2y+z=6

) is incorrect; the correct answer is

y + 2z = 6

because it satisfies the conditions of being perpendicular to the yz-plane and passing through the given points.

2 Medium

2 of 53

Application +4 | -1

\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)

and

\mathrm{D}(-1,-4,-2)

are the vertices of a quadrilateral

\mathrm{ABCD},

then its area is:

View answer and explanation

Correct option: A

12 \sqrt{29}

Option A (

12 \sqrt{29}

) is correct. The area of the quadrilateral is found by dividing it into two triangles and using the cross product to find the area of each triangle.

3 Medium

3 of 53

Application +4 | -1

The equation of the plane through the point

(-1,3,2)

and perpendicular to the planes

x+2 y+2 z=5

and

3 x+3 y+2 z=8

, is

View answer and explanation

Correct option: A

2 x-4 y+3 z+8=0

Option A (

2x-4y+3z+8=0

) is correct because it is the equation of the plane passing through the given point and perpendicular to the two given planes.

4 Medium

4 of 53

Application +4 | -1

If a parallelopiped is formed by planes drawn through the points

(2,3,5)

and

(5,9,7)

parallel to the coordinate plane, then the length of edges and diagonal of parallelopiped are

View answer and explanation

Correct option: B

(3,6,2), 7

units

Option B ((3,6,2), 7 units) is correct because the lengths of the edges are the differences in coordinates, and the length of the diagonal is

\sqrt{{3^2 + 6^2 + 2^2}} = 7

5 Medium

5 of 53

Analyzing +4 | -1

The equation

\sqrt {{{\left( {x - 2} \right)}^2} + {{\left( {y - 1} \right)}^2}} + \sqrt {{{\left( {x + 2} \right)}^2} + {{\left( {y - 4} \right)}^2}} = 5

represents

View answer and explanation

Correct option: C

Line segment

The correct answer is C (Line segment) because the sum of distances from a point to two fixed points equals the distance between the fixed points, which is the definition of a line segment.

6 Medium

6 of 53

Application +4 | -1

The co-ordinates of the point

P

on the line

\mathbf{r}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(-\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})

which is nearest to the origin is

View answer and explanation

Correct option: A

\left(\frac{2}{3}, \frac{4}{3}, \frac{2}{3}\right)

Option A (

\left( \frac{{2}}{{3}}, \frac{{4}}{{3}}, \frac{{2}}{{3}} \right)

) is correct because it is the point on the line closest to the origin, found by minimizing the distance from the origin to a general point on the line.

7 Medium

7 of 53

Application +4 | -1

The equation of the plane through the line

x+y+z+3=0=2 x-y+3 z+1

and parallel to the line

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

View answer and explanation

Correct option: A

x-5 y+3 z=7

Option A (

x-5y+3z=7

) is correct because it is the equation of the plane through the line and parallel to the given line.

8 Medium

8 of 53

Application +4 | -1

Let the point, on the line passing through the points

{P}(1,-2,3)

and

Q(5,-4,7)

, farther from the origin and at a distance of 9 units from the point

P

, be

(\alpha, \beta, \gamma)

. Then

\alpha^2+\beta^2+\gamma^2

is equal to:

View answer and explanation

Correct option: A

155

Option D (

\frac{\pi}{32}

) is correct because, after solving the differential equation and applying the initial condition, y(2) equals

\frac{\pi}{32}

9 Medium

9 of 53

Application +4 | -1

The coordinates of the point where the line segment joining

A(5,1,6)

and

B(3,4,1)

crosses the

Y Z

-plane are

View answer and explanation

Correct option: C

\left(0, \frac{17}{2},-\frac{13}{2}\right)

Option A

\left(0, \frac{{17}}{{2}},-\frac{{13}}{{2}}\right)

is correct because it represents the point where the line segment AB intersects the YZ-plane, found by setting the x-coordinate to zero.

10 Medium

10 of 53

Application +4 | -1

The equation of the plane passing through the straight line

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

and perpendicular to the plane

x+

2 y+z=12

View answer and explanation

Correct option: B

9 x-2 y-5 z+4=0

Option B (

9x-2y-5z+4=0

) is correct because it is the equation of the plane passing through the straight line and perpendicular to the given plane.

11 Medium

11 of 53

Application +4 | -1

The length of the perpendicular from the origin to the plane passing though three non-collinear points a, b, c is

View answer and explanation

Correct option: A

\frac{[\mathbf{a b} \mathbf{c}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{c} \times \mathbf{a}+\mathbf{b} \times \mathbf{c}|}

Option A is correct because it represents the formula for the length of the perpendicular from the origin to the plane passing through three non-collinear points a, b, c.

12 Medium

12 of 53

Application +4 | -1

The shortest distance of the point

(1,2,3)

from

X, Y

and

Z

-axes, respectively are

View answer and explanation

Correct option: D

\sqrt{13}, \sqrt{10}, \sqrt{5}

Option D (

\sqrt{13}, \sqrt{10}, \sqrt{5}

) is correct because the shortest distance from the X, Y, and Z axes are calculated using the formulas

\sqrt{y^2+z^2}

\sqrt{x^2+z^2}

, and

\sqrt{x^2+y^2}

respectively.

13 Medium

13 of 53

Application +4 | -1

The sum of magnitudes of two forces acting at a point is 18 and the magnitude of their resultant is 12 . If the resultant is at

90^{\circ}

with the force of smaller magnitude, then their magnitudes are

View answer and explanation

Correct option: C

5,13

Option C (5,13) is correct because these magnitudes satisfy the given conditions: their sum is 18, their resultant is 12, and the resultant is at 90 degrees with the smaller force.

14 Medium

14 of 53

Application +4 | -1

Let

A(2,3,5)

and

C(-3,4,-2)

be opposite vertices of a parallelogram

A B C D

. If the diagonal

\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}

, then the area of the parallelogram is equal to:

View answer and explanation

Correct option: B

\frac{1}{2} \sqrt{474}

The correct answer is B (

\frac{1}{2} \sqrt{474}

) because it is the area of the parallelogram calculated using the cross product of its diagonals.

15 Medium

15 of 53

Analyzing +4 | -1

Let

\mathrm{O}

be origin and point

P_1(1,1,1)

lie on

L_1=x=y=z

. Projection of

O P_1

on x-axis be

O M_1

. Projection of

O M_1

L_1

O P_2

and so on. Then the value of

\left|P_1 M_1\right|+\left|P_2 M_2\right| \ldots \ldots \infty

View answer and explanation

Correct option: B

\frac{3}{\sqrt{2}}

Option B (

\frac{3}{\sqrt{2}}

) is correct because it represents the sum of an infinite geometric series formed by the projections.

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16 Medium

16 of 53

Application +4 | -1

{A}(3,2,0), {B}(5,3,2), {C}(-9,6,-3)

are three points forming a triangle. If

{AD}

, the bisector of

\angle {BAC}

meets

{BC}

{D}

, then coordinates of

{D}

are

View answer and explanation

Correct option: D

\left(\frac{19}{8}, \frac{57}{16}, \frac{17}{16}\right)

Option D ( (19/8, 57/16, 17/16) ) is correct because it represents the coordinates of point D, which divides BC in the ratio of AB:AC, calculated using the section formula.

17 Medium

17 of 53

Application +4 | -1

What is the length of foot of perpendicular drawn from the point

P(3,4,5)

Y

-axis?

View answer and explanation

Correct option: B

\sqrt{34}

units

Option B (

\sqrt{34}

units) is correct because it represents the distance between the point P(3,4,5) and its projection on the Y-axis, which is (0,4,0).

18 Medium

18 of 53

Application +4 | -1

The point (0, 3) is nearest to the curve

{x^2} = 2y

View answer and explanation

Correct option: C

(2, 2)

Option C (2, 2) is correct because it represents the point on the curve

{x^2} = 2y

that minimizes the distance to the point (0, 3).

19 Medium

19 of 53

Application +4 | -1

The locus of a point

P

which moves such that

2PA = 3PB

, where

A\left( {0,\;0} \right)

and

B\left( {4,\; - 3} \right)

are points, is

View answer and explanation

Correct option: B

5{x^2} + 5{y^2} - 72x + 54y + 225 = 0

Option B (

5{x^2} + 5{y^2} - 72x + 54y + 225 = 0

) is correct because it represents the locus of point P satisfying the condition

2PA = 3PB

, derived by squaring both sides and simplifying the equation.

20 Medium

20 of 53

Application +4 | -1

If the coordinates of orthocentre

O'

and centroid

G

of a

\Delta \;ABC

are

\left( {0,1} \right)

and

\left( {2,3} \right)

respectively, then the coordinates of the circumcentre are

View answer and explanation

Correct option: D

\left( {3,\;4} \right)

Option D

\left( {3,\;4} \right)

is correct because the circumcentre coordinates are found using the section formula with the centroid dividing the line joining orthocentre and circumcentre in the ratio 1:2.

21 Medium

21 of 53

Application +4 | -1

The point on Y-axis, which is at a distance of

\sqrt{10}

units from the point

(1,2,3)

View answer and explanation

Correct option: B

(0,2,0)

Option B ((0,2,0)) is correct because the distance between (1,2,3) and (0,y,0) is

\sqrt{{(1-0)^2 + (2-y)^2 + (3-0)^2}} = \sqrt{{10}}

, which gives

y=2

22 Medium

22 of 53

Application +4 | -1

The angle between the lines whose direction cosines are given by the equations

3 l+{m}+5 {n}=0,6 {nm}-2 {n} l+5 l {m}=0

View answer and explanation

Correct option: B

\cos ^{-1}\left(-\frac{1}{6}\right)

Option B,

\cos ^{-1}\left(-\frac{1}{6}\right)

, is the correct answer because it is the angle calculated using the direction cosines derived from the given equations.

23 Medium

23 of 53

Application +4 | -1

The equation of the locus of a point equidistant from the points

\left( {{a_1},\;{b_1}} \right)

and

\left( {{a_2},\;{b_2}} \right)

\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0

, then the value of

c

View answer and explanation

Correct option: C

\frac{1}{2}\left( {a_2^2 + b_2^2 - a_1^2 - b_1^2} \right)

Option C (

\frac{1}{2}\left( {a_2^2 + b_2^2 - a_1^2 - b_1^2} \right)

) is correct because it represents the constant term in the equation of the locus of a point equidistant from two given points.

24 Medium

24 of 53

Application +4 | -1

A(6,3,2), B(5,1,4), C(3,-4,7), D(0,2,5)

are four points then length of projection of

C D

A B

View answer and explanation

Correct option: A

\frac{13}{3}

Option A (

\frac{13}{3}

) is correct because it is the length of the projection of vector CD onto vector AB, calculated using the dot product and magnitude.

25 Medium

25 of 53

Application +4 | -1

If the line joining

A(1,3,4)

and

B

is divided by the point

(-2,3,5)

in the ratio

1: 3

, then

B

View answer and explanation

Correct option: A

(-11,3,8)

Option A ((-11,3,8)) is correct because it is found by using the section formula to determine the coordinates of point B.

26 Medium

26 of 53

Application +4 | -1

The shortest distance (S.D) between lines

\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}

and

\frac{x+3}{-3}-\frac{y+7}{2}=\frac{z-6}{4}

View answer and explanation

Correct option: A

3 \sqrt{30}

Option A (

3 \sqrt{30}

) is correct because it represents the shortest distance between the two given lines, calculated by finding a common perpendicular.

27 Medium

27 of 53

Application +4 | -1

The line

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

intervals the curve

x y=k^2, z=0 \mathrm{x}

then

k=

View answer and explanation

Correct option: B

\pm 5

Option B (

\pm 5

) is correct because substituting the parametric form of the line into the curve equation yields k =

\pm 5

28 Medium

28 of 53

Application +4 | -1

The ratio in which

Z X

-plane divides the line segment

A B

joining the points

A(4,2,3)

and

B(-2,4,5)

is equal to

View answer and explanation

Correct option: B

1: 2

, externally

Option B (

1: 2

, externally) is correct because the ZX-plane divides the line segment in the ratio

-y_1 : y_2

, which is negative, indicating external division.

29 Medium

29 of 53

Analyzing +4 | -1

A B C D

is a tetrahedron such that each

\triangle A B C, \triangle A B D

and

\triangle A C D

has a right angle at

A

. If

\operatorname{ar}(\triangle A B C)=k_1

\operatorname{ar}(\triangle A B D)=k_2, \operatorname{ar}(\triangle B C D)=k_3

, then

\operatorname{ar}(\triangle A C D)

View answer and explanation

Correct option: C

\sqrt{\left|k_1^2+k_2^2-k_3^2\right|}

The correct answer is C (12) because it is the value of 6(a+b+c) after finding the relationship between a, b, and c from the coplanarity conditions.

30 Medium

30 of 53

Application +4 | -1

The ratio in which the line joining

(1,2,3)

and

(-3,4,-5)

is divided by

X Y

-plane, is

View answer and explanation

Correct option: B

3: 5

Option B (3: 5) is correct because the XY-plane divides the line joining (1,2,3) and (-3,4,-5) in the ratio k:1, where the z-coordinate of the point of division is 0, giving

k = \frac{{3}}{{5}}

, so the ratio is 3:5.

Revision notes and key takeaways

Review 3D GEOMETRY SYSTEM in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 53 single-correct-answer MCQs on 3D GEOMETRY SYSTEM.
✓ Try each question before opening its answer. Read the explanation where one is available.
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