RD SHARMA SOLUTION CHAPTER- 28 Introduction to 3D Coordinate Geometry I CLASS 11TH MATHEMATICS-EDUGROW

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

Question 1(i)

Name the octants in which the following points lie:

 (i) (5, 2, 3)Solution 1(i)

All are positive, so octant is XOYZQuestion 1(ii)

Name the octants in which the following points lie:

(ii) (-5, 4, 3)Solution 1(ii)

X is negative and rest are positive, so octant is X^‘OYZQuestion 1(iii)

Name the octants in which the following points lie:

(4, -3, 5)Solution 1(iii)

Y is negative and rest are positive, so octant is XOY^‘ZQuestion 1(iv)

Name the octants in which the following points lie:

(7, 4, -3)Solution 1(iv)

Z is negative and rest are positive, so octant is XOYZ^‘Question 1(v)

Name the octants in which the following points lie:

(-5, -4, 7)Solution 1(v)

X and Y are negative and Z is positive, so octant is X’OY’ZQuestion 1(vi)

Name the octants in which the following points lie:

(-5, -3, -2)Solution 1(vi)

All are negative, so octant is X^‘OY^‘Z^‘Question 1(vii)

Name the octants in which the following points lie:

(2, -5, -7)Solution 1(vii)

Y and Z are negative, so octant is XOY^‘Z^‘Question 1(viii)

Name the octants in which the following points lie:

(-7, 2, -5)Solution 1(viii)

X and Z are negative, so octant is X^‘OYZ^‘Question 2(i)

Find the image of :

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

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

Find the image of :

(-5, 4, -3) in the xz-plane. Solution 2(ii)

XZ plane is y-axis, so sign of y will be changed. So answer is (-5, -4, -3)Question 2(iii)

Find the image of :

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

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

Find the image of :

(-5, 0, 3) in the xz-plane Solution 2(iv)

XZ plane is y-axis, so sign of Y will change, So answer is (-5, 0, 3)Question 2(v)

Find the image of :

(-4, 0, 0) in the xy-plane Solution 2(v)

XY plane is Z-axis, so sign of Z will change So answer is (-4, 0, 0)Question 3

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 3

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 4

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 4

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

5, 5, 5 are lengths of edgesQuestion 5

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 5

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

2, 2, 3 are lengths of edgesQuestion 6

Find the distances of the point p(-4, 3, 5) from the coordinate axes. Solution 6

Question 7

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 7

(-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(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

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(i)

Solution 20(i)

Question 20(ii)

Solution 20(ii)

Question 20(iii)

Solution 20(iii)

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 20(iv)

Verify the following

(5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.Solution 20(iv)

Question 24

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 24

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