# Chapter 28 : Introduction to 3-D coordinate geometry - Rd Sharma Solutions for Class 11-science Maths CBSE

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## 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 X'OYZ

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 XOY'Z

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 X'OY'Z'

Question 7

Name the octants in which the following points lie:

(2, -5, -7)

Solution 7

Y and Z are negative, so octant is XOY'Z'

Question 8

Name the octants in which the following points lie:

(-7, 2, -5)

Solution 8

X and Z are negative, so octant is X'OYZ'

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 Solution 27 Question 28 Solution 28 Question 29 Solution 29  Question 30

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 30 Question 31

Verify the following

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

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 