# RD Sharma Solution CLass 11 Mathematics Chapter 28 Introduction to 3D Coordinate Geometry

## Chapter 28 – Introduction to 3-D coordinate geometry Exercise Ex. 28.1

Question 1

Name the octants in which the following points lie:

(i) (5, 2, 3)

Solution 1

All are positive, so octant is XOYZ

Question 2

Name the octants in which the following points lie:

(ii)
(-5, 4, 3)

Solution 2

X is negative and rest are
positive, so octant is XOYZ

Question 3

Name the octants in which the following points lie:

(4,
-3, 5)

Solution 3

Y is negative and rest are
positive, so octant is XOYZ

Question 4

Name the octants in which the following points lie:

(7,
4, -3)

Solution 4

Z is negative and rest are
positive, so octant is XOYZ

Question 5

Name the octants in which the following points lie:

(-5,
-4, 7)

Solution 5

X and Y are negative and Z
is positive, so octant is X’OY’Z

Question 6

Name the octants in which the following points lie:

(-5,
-3, -2)

Solution 6

All are negative, so
octant is XOYZ

Question 7

Name the octants in which the following points lie:

(2,
-5, -7)

Solution 7

Y and Z are negative, so
octant is XOYZ

Question 8

Name the octants in which the following points lie:

(-7,
2, -5)

Solution 8

X and Z are negative, so
octant is XOYZ

Question 9

Find the image of :

(-2, 3, 4) in the yz-plane

Solution 9

YZ plane is x-axis, so
sign of x will be changed. So answer is (2, 3, 4)

Question 10

Find the image of :

(-5, 4, -3) in the xz-plane.

Solution 10

XZ plane is y-axis, so
sign of y will be changed. So answer is (-5, -4, -3)

Question 11

Find the image of :

(5, 2, -7) in the xy-plane

Solution 11

XY-plane is z-axis, so
sign of Z will change. So answer is (5, 2, 7)

Question 12

Find the image of :

(-5, 0, 3) in the xz-plane

Solution 12

XZ plane is y-axis, so
sign of Y will change, So answer is (-5, 0, 3)

Question 13

Find the image of :

(-4, 0, 0) in the xy-plane

Solution 13

XY plane is Z-axis, so
sign of Z will change So answer is (-4, 0, 0)

Question 14

A
cube of side 5 has one vertex at the point (1, 0, -1), and the three edges
from this vertex are, respectively, parallel to the negative x and y axes and
positive z-axis. Find the value coordinates of the other vertices of the
cube.

Solution 14

Vertices of cube are

(1, 0, -1) (1, 0, 4) (1,
-5, -1)

(1, -5, 4) (-4, 0, -1)
(-4, -5, -4)

(-4, -5, -1) (4, 0, 4) (1,
0, 4)

Question 15

Planes
are drawn parallel to the coordinate planes through the points (3, 0, -1) and
(-2, 5, 4). Find the lengths of the edges of the parallelepiped so formed.

Solution 15

3-(-2)=5, |0-5|=5,
|-1-4|=5

5, 5, 5 are lengths of
edges

Question 16

Planes
are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the
coordinate planes. Find the lengths of the edges of the rectangular
parallelepiped so formed.

Solution 16

5-3=2, 0-(-2)=2, 5-2=3

2, 2, 3 are lengths of
edges

Question 17

Find
the distances of the point p(-4, 3, 5) from the
coordinate axes.

Solution 17

Question 18

The
coordinate of a point are (3, -2, 5). Write down the
coordinates of seven points such that the absolute values of their
coordinates are the same as those of the coordinates of the given point.

Solution 18

(-3, -2, -5) (-3, -2, 5)
(3, -2, -5) (-3, 2, -5) (3, 2, 5)

(3, 2, -5) (-3, 2, 5)

## Chapter 28 – Introduction to 3-D coordinate geometry Exercise Ex. 28.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Verify the following

(5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are
the vertices of a rhombus.

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.

Solution 31

## Chapter 28 – Introduction to 3-D coordinate geometry Exercise Ex. 28.3

Question 1

The
vertices of the triangle are A(5, 4, 6), B(1, -1, 3)
and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the
coordinates of D and the Length AD.

Solution 1

Question 2

A
point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find its coordinates.

Solution 2

Question 3

Show
that three points A(2, 3, 4), B(-1, 2, -3) and C(-4, 1, -10) are collinear
and find the ratio in which C divides AB.

Solution 3

Question 4

Find
the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the
yz-plane.

Solution 4

Question 5

Find the ratio in which the line segment joining the
points (2, -1, 3) and (-1, 2, 1) is divided by the plane

x+ y +
z = 5.

Solution 5

Question 6

If
the points A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) are collinear, find the
ratio in which C divides AB.

Solution 6

Question 7

The
mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C.

Solution 7

Question 8

A(1, 2, 3),
B(0, 4, 1), C(-1, -1, -3) are the vertices of a triangle ABC. Find the point
in which the bisector of the angle  meets BC.

Solution 8

Question 9

Find
the ratio in which the sphere x2+y2 +z2 =
504 divides the line joining the points (12, -4, 8) and (27, -9, 18).

Solution 9

Question 10

Show that the plane ax + by + cz
+ d = 0 divides the line joining the points (x1,y1,z1)
and (x2,y2,z2)
in the ratio –

Solution 10

Question 11

Find
the centroid of a triangle, mid-points of whose
sides are (1, 2, -3), (3, 0, 1) and (-1, 1, -4).

Solution 11

Question 12

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinate of A and B are (3, -5, 7) and (-1,
7, -6) respectively, find the coordinates of the point C.

Solution 12

Question 13

Find
the coordinates of the points which tisect the line
segment joining the points P(4, 2, -6) and Q(10,
-16, 6).

Solution 13

Question 14

Using
section formula, show that the points A(2, -3, 4),
B(-1, 2, 1) and C(0, 1/3, 2) are collinear.

Solution 14

Question 15

Given
that P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are
collinear. Find the ratio in which Q divides PR.

Solution 15

Question 16

Find
the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the yz-plane.

Solution 16

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