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# RD Sharma solutions for Class 11 Mathematics chapter 28 - Introduction to three dimensional coordinate geometry

## Chapter 28 : Introduction to three dimensional coordinate geometry

#### Pages 6 - 7

Q 1.2 | Page 6

Name the octants in which the following points lie:

(–5, 4, 3)

Q 1.3 | Page 6

Name the octants in which the following points lie:

(4, –3, 5)

Q 1.4 | Page 6

Name the octants in which the following points lie:

(7, 4, –3)

Q 1.5 | Page 6

Name the octants in which the following points lie:

(–5, –4, 7)

Q 1.6 | Page 6

Name the octants in which the following points lie:

(–5, –3, –2)

Q 1.7 | Page 6

Name the octants in which the following points lie:

(2, –5, –7)

Q 1.8 | Page 6

Name the octants in which the following points lie:

(–7, 2 – 5)

Q 2.1 | Page 6

Find the image  of:

(–2, 3, 4) in the yz-plane.

Q 2.2 | Page 6

Find the image  of:

(–5, 4, –3) in the xz-plane.

Q 2.3 | Page 6

Find the image  of:

(5, 2, –7) in the xy-plane.

Q 2.4 | Page 6

Find the image  of:

(–5, 0, 3) in the xz-plane.

Q 2.5 | Page 6

Find the image  of:

(–4, 0, 0) in the xy-plane.

Q 3 | Page 6

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 coordinates of the other vertices of the cube.

Q 4 | Page 6

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.

Q 5 | Page 7

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.

Q 6 | Page 7

Find the distances of the point P(–4, 3, 5) from the coordinate axes.

Q 7 | Page 7

The coordinates 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.

#### Pages 9 - 10

Q 1.1 | Page 9

Find the distance between the following pairs of points:

P(1, –1, 0) and Q(2, 1, 2)

Q 1.2 | Page 9

Find the distance between the following pairs of point:

A(3, 2, –1) and B(–1, –1, –1).

Q 2 | Page 9

Find the distance between the points P and Q having coordinates (–2, 3, 1) and (2, 1, 2).

Q 2.1 | Page 10

Verify the following:

(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.

Q 3.1 | Page 9

Using distance formula prove that the following points are collinear:

A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)

Q 3.2 | Page 9

Using distance formula prove that the following points are collinear:

P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)

Q 3.3 | Page 9

Using distance formula prove that the following points are collinear:

A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)

Q 4.1 | Page 9

Determine the points in xy-plan are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).

Q 4.2 | Page 9

Determine the points in yz-plane and are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).

Q 4.3 | Page 9

Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).

Q 5 | Page 9

Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).

Q 6 | Page 9

Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).

Q 7 | Page 9

Find the points on z-axis which are at a distance $\sqrt{21}$from the point (1, 2, 3).

Q 8 | Page 9

Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.

Q 9 | Page 10

Show that the points (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of an isosceles right-angled triangle.

Q 10 | Page 9

Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of a square.

Q 11 | Page 10

Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.

Q 12 | Page 10

Show that the points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) are the vertices of a rhombus.

Q 13 | Page 10

Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.

Q 14 | Page 10

Show that the points (3, 2, 2), (–1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.

Q 15 | Page 10

Find the coordinates of the point which is equidistant  from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).

Q 16 | Page 10

If A(–2, 2, 3) and B(13, –3, 13) are two points.
Find the locus of a point P which moves in such a way the 3PA = 2PB.

Q 17 | Page 10

Find the locus of P if PA2 + PB2 = 2k2, where A and B are the points (3, 4, 5) and (–1, 3, –7).

Q 18 | Page 10

Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.

Q 19 | Page 10

Are the points A(3, 6, 9), B(10, 20, 30) and C(25, –41, 5), the vertices of a right-angled triangle?

Q 20.1 | Page 10

Verify the following:

(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.

Q 20.2 | Page 10

Verify the following:

(0, 7, 10), (–1, 6, 6) and (–4, 9, –6) are vertices of a right-angled triangle.

Q 20.3 | Page 10

Verify the following:

(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are vertices of a parallelogram.

Q 20.4 | Page 10

Verify the following:

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

Q 21 | Page 10

Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).

Q 22 | Page 10

Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.

Q 23 | Page 10

Show that the points A(1, 2, 3), B(–1, –2, –1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram ABCD, but not a rectangle.

Q 24 | Page 10

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.

#### Pages 19 - 20

Q 1 | Page 19

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.

Q 2 | Page 19

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.

Q 3 | Page 20

Show that the 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

Q 4 | Page 20

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

Q 5 | Page 20

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.

Q 6 | Page 20

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

Q 7 | Page 20

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 AB and C.

Q 8 | Page 20

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 ∠BAC meets BC.

Q 9 | Page 20

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

Q 10 | Page 20

Show that the plane ax + by cz + d = 0 divides the line joining the points (x1y1z1) and (x2y2z2) in the ratio $- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}$

Q 11 | Page 20

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

Q 12 | Page 20

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

Q 13 | Page 20

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

Q 14 | Page 20

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

Q 15 | Page 20

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

Q 16 | Page 20

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

#### Page 22

Q 1 | Page 22

Write the distance of the point P (2, 3,5) from the xy-plane.

Q 2 | Page 22

Write the distance of the point P(3, 4, 5) from z-axis.

Q 3 | Page 22

If the distance between the points P(a, 2, 1) and Q (1, −1, 1) is 5 units, find the value of a

Q 4 | Page 22

The coordinates of the mid-points of sides AB, BC and CA of  △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.

Q 5 | Page 22

Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.

Q 6 | Page 22

Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.

Q 7 | Page 22

Write the coordinates of third vertex of a triangle having centroid at the origin and two vertices at (3, −5, 7) and (3, 0, 1).

Q 8 | Page 22

What is the locus of a point for which y = 0, z = 0?

Q 9 | Page 22

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

Q 10 | Page 22

Find the point on y-axis which is at a distance of  $\sqrt{10}$ units from the point (1, 2, 3).

Q 11 | Page 22

Find the point on x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).

Q 12 | Page 22

Find the coordinates of a point equidistant from the origin and points A (a, 0, 0), B (0, b, 0) andC(0, 0, c).

Q 13 | Page 22

Write the coordinates of the point P which is five-sixth of the way from A(−2, 0, 6) to B(10, −6, −12).

Q 14 | Page 22

If a parallelopiped is formed by the planes drawn through the points (2,3,5) and (5, 9, 7) parallel to the coordinate planes, then write the lengths of edges of the parallelopiped and length of the diagonal.

Q 15 | Page 22

Determine the point on yz-plane which is equidistant from points A(2, 0, 3), B(0, 3,2) and C(0, 0,1).

Q 16 | Page 22

If the origin is the centroid of a triangle ABC having vertices A(a, 1, 3), B(−2, b −5) and C (4, 7, c), find the values of a, b, c.

#### Pages 22 - 23

Q 1 | Page 22

The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is

(a) 2 : 3

(b) 3 : 2

(c) –2 : 3

(d) 4 : –3

Q 2 | Page 22

The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is

(a) a : b

(b) b : c

(c) c a

(d) c : b

Q 3 | Page 22

If P(0, 1, 2), Q(4, –2, 1) and O(0, 0, 0) are three points, then ∠POQ

(a) $\frac{\pi}{6}$

(b) $\frac{\pi}{4}$

(c)$\frac{\pi}{3}$

(d)$\frac{\pi}{2}$

Q 4 | Page 22

If the extremities of the diagonal of a square are (1, –2, 3 and (2, –3, 5), then the length of the side is

(a)$\sqrt{6}$

(b) $\sqrt{3}$

(c) $\sqrt{5}$

(d) $\sqrt{7}$

Q 5 | Page 23

The points (5, –4, 2), (4, –3, 1), (7, 6, 4) and (8, –7, 5) are the vertices of

(a) a rectangle

(b) a square

(c) a parallelogram

(d) none of these

Q 6 | Page 23

In a three dimensional space the equation x2 – 5x + 6 = 0 represents

(a) points

(b) planes

(c) curves

(d) pair of straight lines

Q 7 | Page 23

Let (3, 4, –1) and (–1, 2, 3) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to

(a) 2

(b) 3

(c) 6

(d) 7

Q 8 | Page 23

XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio

(a) 3 : 7

(b) 2 : 7

(c) –3 : 7

(d) –2 : 7

Q 9 | Page 23

What is the locus of a point for which y = 0, z = 0?

(a) - axis

(b) y - axis

(c) z - axis

(d) yz - plane

Q 10 | Page 23

The coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are

(a) (3, 4, 0)

(b) (0, 4, 5)

(c) (3, 0, 5)

(d) (3, 0, 0)

Q 11 | Page 23

The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are

(a) (6, 0, 0)

(b) (0, 7, 0)

(c) (0, 0, 8)

(d) (0, 7, 8)

We know that the y  and z coordinates on x - axis are 0
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are (6, 0, 0)
Hence, the correct answer is option (a).

Q 12 | Page 23

The perpendicular distance of the point P (6, 7, 8) from xy - plane is

(a) 8

(b) 7

(c) 6

(d) 10

Q 13 | Page 23

The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is

(a) 10

(b)$\sqrt{34}$

(c) $\sqrt{113}$

(d) 512

Q 14 | Page 23

The perpendicular distance of the point P(3, 3,4) from the x-axis is

(a)$3\sqrt{2}$

(b) 5

(c) 3

(d) 4

Q 15 | Page 23

The length of the perpendicular drawn from the point P(a, b, c) from z-axis is

(a)$\sqrt{a^2 + b^2}$

(b) $\sqrt{b^2 + c^2}$

(c)$\sqrt{a^2 + c^2}$

(d) $\sqrt{a^2 + b^2 + c^2}$

## RD Sharma solutions for Class 11 Mathematics chapter 28 - Introduction to three dimensional coordinate geometry

RD Sharma solutions for Class 11 Maths chapter 28 (Introduction to three dimensional coordinate geometry) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 28 Introduction to three dimensional coordinate geometry are Three Dimessional Space, Section Formula, Distance Between Two Points, Coordinates of a Point in Space, Coordinate Axes and Coordinate Planes in Three Dimensional Space.

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