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

text Let    A end text equals open parentheses 0 comma 7 comma 10 close parentheses text ,  B = end text open parentheses minus 1 comma 6 comma 6 close parentheses text   and end text space C equals open parentheses minus 4 comma 9 comma 6 close parentheses

A B equals square root of left parenthesis 0 plus 1 right parenthesis squared plus left parenthesis 7 minus 6 right parenthesis squared plus left parenthesis 10 minus 6 right parenthesis squared end root
equals square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root
equals square root of 18
equals 3 square root of 2 space space text units end text

B C equals square root of left parenthesis minus 1 plus 4 right parenthesis squared plus left parenthesis 6 minus 9 right parenthesis squared plus left parenthesis 6 minus 6 right parenthesis squared end root
equals square root of left parenthesis 3 right parenthesis squared plus left parenthesis 3 right parenthesis squared plus 0 end root
equals square root of 18
equals 3 square root of 2 space space text units end text

A C equals square root of left parenthesis 0 plus 4 right parenthesis squared plus left parenthesis 7 minus 9 right parenthesis squared plus left parenthesis 10 minus 6 right parenthesis squared end root
equals square root of left parenthesis 4 right parenthesis squared plus left parenthesis minus 2 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root
equals square root of 36
equals 6 space space text units end text

left parenthesis A B right parenthesis squared plus left parenthesis B C right parenthesis squared
equals open parentheses 3 square root of 2 close parentheses squared plus open parentheses 3 square root of 2 close parentheses squared
equals 18 plus 18
equals 36
equals left parenthesis A C right parenthesis squared

text Also   end text l left parenthesis A B right parenthesis equals l left parenthesis B C right parenthesis

text Hence    end text open parentheses 0 comma 7 comma 10 close parentheses text ,  end text open parentheses minus 1 comma 6 comma 6 close parentheses text   and end text space open parentheses minus 4 comma 9 comma 6 close parentheses space text are   the   vertices   of   an   isosceles   right-angled   triangle. end text

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

 

 

 

 

 

 

 

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