If $A\left( {6, - 3} \right),B\left( { - 3,5} \right),C\left( {4, - 2} \right),P\left( {{\rm{\alpha }},{\rm{\beta }}} \right),$ then the ratio of the areas of the triangles $PBC,ABC$ is

$\left| {{\rm{\alpha }} + {\rm{\beta }} - 2} \right|$

$\left| {{\rm{\alpha }} + {\rm{\beta }}} \right|$

$\left| {{\rm{\alpha }} - {\rm{\beta }}} \right|$

$\left| {{\rm{\alpha }} + {\rm{\beta }} + 2} \right|$

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JEE • MATHEMATICS • 3D GEOMETRY SYSTEM

3d geometry system practice · Easy

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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 85 questions

1 Easy

1 of 85

Application +4 | -1

The mid point of the line joining the points

\left( { - 10,\;8} \right)

and

\left( { - 6,\;12} \right)

divides the line joining the points

\left( {4,\; - 2} \right)

and

\left( { - 2,\;4} \right)

in the ratio

View answer and explanation

Correct option: D

2:1

externally

Option D (

2:1

externally) is correct because the midpoint divides the line externally in the ratio 2:1, as calculated using the section formula.

2 Easy

2 of 85

Application +4 | -1

The value of

p

for which the lines

2 x+y-3=0

3 x-y-2=0

and

x-p y+5=0

may intersect at a point is

View answer and explanation

Correct option: D

Option D (6) is correct because the lines intersect at a single point only when p=6, which is found by solving the system of equations.

3 Easy

3 of 85

Analyzing +4 | -1

If the vertices of a triangle at

O\left( {0,0} \right),A\left( {a,0} \right)

and

B\left( {0,a} \right).

Then, the distance between its circumcentre and orthocentre is

View answer and explanation

Correct option: B

\frac{a}{{\sqrt 2 }}

Option B (

\frac{a}{{\sqrt 2 }}

) is correct because the orthocenter is at the origin and the circumcenter is the midpoint of the hypotenuse, allowing us to calculate the distance between them.

4 Easy

4 of 85

Application +4 | -1

The coordinates of the foot of the perpendicular drawn from the point

A(1,0,3)

to the join of the points

B(4,7,1)

and

C(3,5,3)

are

View answer and explanation

Correct option: A

\left(\frac{5}{3}, \frac{7}{3}, \frac{17}{3}\right)

Option A (

\left(\frac{5}{3}, \frac{7}{3}, \frac{17}{3}\right)

) is correct because it represents the coordinates of the foot of the perpendicular calculated using section formula and dot product.

5 Easy

5 of 85

Analyzing +4 | -1

The area of the triangle formed by the three lines

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

and

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

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the area of the triangle formed by the given lines is calculated using the coordinates of the vertices.

6 Easy

6 of 85

Application +4 | -1

The direction cosines of the line which is perpendicular to the lines whose direction cosines are proportional to

(1,-1,2)

and

(2,1,-1)

, are

View answer and explanation

Correct option: B

-\frac{1}{\sqrt{35}}, \frac{5}{\sqrt{35}}, \frac{3}{\sqrt{35}}

Option B (

-\frac{{1}}{{\sqrt{{35}}}}, \frac{{5}}{{\sqrt{{35}}}}, \frac{{3}}{{\sqrt{{35}}}}

) is correct because it represents the direction cosines of a line perpendicular to the two given lines.

7 Easy

7 of 85

Application +4 | -1

Equation of the line passing through

(1,1,1)

and parallel to the plane

2 x+3 y+z+5=0

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the direction ratios of the line are perpendicular to the normal vector of the plane.

8 Easy

8 of 85

Application +4 | -1

If the origin is the centroid of the triangle with vertices

{A}(2 {a}, 2,6), {B}(-4,3 {b},-10)

and

{C}(8,14,2 {c})

, then the sum of value of

a

and

c

View answer and explanation

Correct option: A

Option A (0) is correct because the sum of a and c is 0, calculated using the centroid formula.

9 Easy

9 of 85

Application +4 | -1

The locus of a point which moves so that the difference of the squares of its distances from two given points is constant, is a

View answer and explanation

Correct option: B

plane

The correct answer is B (plane) because the locus results in a linear equation in terms of the position vector.

10 Easy

10 of 85

Application +4 | -1

The line passing through the points

(5,1 a)

, and

(3, b, 1)

crosses the

Y Z

-plane at the point

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

. Then,

View answer and explanation

Correct option: D

a=6, b=4

Option D (

a=6, b=4

) is correct because these are the values of a and b for which the line passing through the given points crosses the YZ-plane at the specified point, found by setting up the equation of the line and solving for a and b.

11 Easy

11 of 85

Application +4 | -1

Ratio in which the

z x

-plane divides the join of

(1,2,3)

and

(4,2,1)

View answer and explanation

Correct option: B

1: 1

externally

Option B is correct because the zx-plane divides the join externally in the ratio found by setting the y-coordinate of the division point to zero.

12 Easy

12 of 85

Application +4 | -1

The ratio in which the plane

{r} \cdot(\hat{i}-2 \hat{j}+2 \hat{k})=17

divides the line joining the points

-2 \hat{i}+4 \hat{j}+7 \hat{k}

and

3 \hat{i}-5 \hat{j}+8 \hat{k}

View answer and explanation

Correct option: C

3: 10

Option C (

3: 10

) is correct because it represents the ratio in which the plane divides the line segment, calculated using the section formula and the plane's equation.

13 Easy

13 of 85

Application +4 | -1

Point

\left( {\frac{1}{2},\; - \frac{{13}}{4}} \right)

divides the line joining the points

\left( {3,\; - 5} \right)

and

\left( { - 7,\;2} \right)

in the ratio of

View answer and explanation

Correct option: A

1:3

internally

Option A (

1:3

internally) is correct because it is the ratio in which the point divides the line segment, calculated using the section formula.

14 Easy

14 of 85

Understanding +4 | -1

The locus represented by

x y+y z-0

View answer and explanation

Correct option: D

a pair of perpendicular planes

Option D (a pair of perpendicular planes) is correct because the equation can be factored into y(x+z) = 0, representing two planes y=0 and x+z=0, which are perpendicular.

15 Easy

15 of 85

Application +4 | -1

Let

P

be the point (1, 0) and

{\cal Q}

be the point on

{y^2} = 8x

. The locus of mid point of

P{\cal Q}

View answer and explanation

Correct option: D

{y^2} - 4x + 2 = 0

Option D

{y^2} - 4x + 2 = 0

is correct because it is the locus of the midpoint derived from the given parabola equation.

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16 Easy

16 of 85

Application +4 | -1

A\left( {6, - 3} \right),B\left( { - 3,5} \right),C\left( {4, - 2} \right),P\left( {\mathrm{\alpha },\mathrm{\beta }} \right),

then the ratio of the areas of the triangles

PBC,ABC

View answer and explanation

Correct option: D

\left| {\mathrm{\alpha } + \mathrm{\beta } - 2} \right|

Option D is correct because the ratio of areas of triangles PBC and ABC is found by calculating the determinants and simplifying to |α + β - 2|.

17 Easy

17 of 85

Application +4 | -1

The locus of a point

P

which moves such that

2PA = 3PB

, where coordinates of points

A

and

B

are (0,0) and (4,-

- 3

), 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 given distance condition.

18 Easy

18 of 85

Understanding +4 | -1

The direction cosines of any normal to the

x y

-plane are

View answer and explanation

Correct option: D

0,0,1

Option D (0,0,1) is correct because the normal vector to the xy-plane is along the z-axis, and its direction cosines are (0,0,1).

19 Easy

19 of 85

Application +4 | -1

The point

P

is equidistant from

A\left( {1,\;3} \right),\;B\left( { - 3,\;5} \right)

and

C\left( {5,\; - 1} \right)

, then

PA

is equal to

View answer and explanation

Correct option: D

5\sqrt {10}

Option D (

5\sqrt {10}

) is correct because it is the distance PA calculated after finding the coordinates of point P, which is equidistant from A, B, and C.

20 Easy

20 of 85

Analyzing +4 | -1

Let A be a fixed point (0, 6) and

B

be a moving point (2t, 0). Let

M

be the mid-point of

AB

and the perpendicular bisectors of

AB

meets the

y

-axis at

C

.
The locus of the mid-point

P

MC

View answer and explanation

Correct option: C

2{x^2} + 3y - 9 = 0

Option C (

2{x^2} + 3y - 9 = 0

) is correct because it represents the locus of the midpoint P of MC, derived by finding the coordinates of M and C and then eliminating the parameter t.

21 Easy

21 of 85

Application +4 | -1

The equation of the plane which contains the origin and the line of intersection of the planes

{r} \cdot {a}=p

and

{r} \cdot {b}=

q

View answer and explanation

Correct option: D

{r} \cdot(q {a}-p {b})=0

Option D (

{r} \cdot(q {a}-p {b})=0

) is correct because it represents the equation of a plane passing through the origin and the line of intersection of the given planes.

22 Easy

22 of 85

Application +4 | -1

If the line through the points

(4,1,2)

and

(5, k, 0)

is parallel to the line through the points

(2,1,1)

and

(3

3,-1)

, then

k

is equal to

View answer and explanation

Correct option: A

Option A (3) is correct because the lines are parallel, so their direction ratios are proportional.

23 Easy

23 of 85

Application +4 | -1

The point moves such that the area of the triangle formed by it with the points (1, 5) and (3,-7) is 21 sq unit. The locus of the point is

View answer and explanation

Correct option: A

6x + y - 32=0

Option A (

6x + y - 32=0

) is correct because it represents the locus of the point derived from the area of the triangle formed by the given points.

24 Easy

24 of 85

Application +4 | -1

P(3,2,-4), Q(5,4,-6)

and

R(9,8,-10)

are collinear, then

R

divides

P Q

in the ratio

View answer and explanation

Correct option: B

3: 2

externally

Option B is correct because R divides PQ externally in the ratio found using the section formula and equating the coordinates.

25 Easy

25 of 85

Application +4 | -1

The direction ratios of the diagonals of a cube which joins the origin to the opposite corner are (when the three concurrent edges of the cube are coordinate axes)

View answer and explanation

Correct option: B

1,1,1

Option B (

1,1,1

) is correct because the direction ratios of the diagonal of a cube joining the origin to the opposite corner are equal to the coordinates of that corner.

26 Easy

26 of 85

Application +4 | -1

If orthocentre and circumcentre of triangle are respectively (1, 1) and (3, 2), then the coordinates of its centroid are

View answer and explanation

Correct option: A

\left( {\frac{7}{3},\frac{5}{3}} \right)

Option A,

\left( {\frac{7}{3},\frac{5}{3}} \right)

, is correct because the centroid divides the line segment joining the orthocenter and circumcenter in the ratio 2:1.

27 Easy

27 of 85

Understanding +4 | -1

{r}

be position vector of any point on a sphere and a and

{b}

are respectively position vectors of the extremities of a diameter, then

View answer and explanation

Correct option: D

(\mathbf{r}-\mathbf{a}) \cdot(\mathbf{r}-\mathbf{b})=0

Option D ((\mathbf{r}-\mathbf{a}) ·(\mathbf{r}-\mathbf{b})=0) is correct because the angle in a semicircle is a right angle, implying the dot product of the vectors representing the sides is zero.

28 Easy

28 of 85

Application +4 | -1

Let

A

be the image of

(2,-1)

with respect to

Y

-axis. Without transforming the origin, coordinate axis are turned at an angle

45^{\circ}

in the clockwise direction. Then, the coordiates of

A

in the new system are

View answer and explanation

Correct option: A

\left(-\frac{1}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right)

Option A

\left(-\frac{1}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right)

is correct because it represents the coordinates of the image of the point after rotation.

29 Easy

29 of 85

Understanding +4 | -1

If a line makes

\alpha, \beta, \gamma

angles with the positive directions of the axes, then

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

is equal to

View answer and explanation

Correct option: B

Option B (2) is correct because

\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2

given that

\alpha, \beta, \gamma

are the angles a line makes with the positive directions of the axes.

30 Easy

30 of 85

Understanding +4 | -1

A

and

B

are two fixed points, then the locus of a point which moves in such a way that the angle,

APB

is a right angle is

View answer and explanation

Correct option: A

A circle

Option A is correct because the locus of a point P such that angle APB is a right angle, where A and B are fixed points, is a circle with AB as the diameter.

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 85 single-correct-answer MCQs on 3D GEOMETRY SYSTEM.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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