# NCERT solutions for Class 11 Mathematics chapter 12 - Introduction to Three Dimensional Geometry [Latest edition]

#### Chapters ## Chapter 12: Introduction to Three Dimensional Geometry

Exercise 12.1Exercise 12.2Exercise 12.3Miscellaneous Exercise
Exercise 12.1 [Page 271]

### NCERT solutions for Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry Exercise 12.1 [Page 271]

Exercise 12.1 | Q 1 | Page 271

A point is on the x-axis. What are its y-coordinates and z-coordinates?

Exercise 12.1 | Q 2 | Page 271

A point is in the XZ-plane. What can you say about its y-coordinate?

Exercise 12.1 | Q 3 | Page 271

Name the octants in which the following points lie:

(1, 2, 3), (4, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),

(–3, –1, 6), (2, –4, –7)

Exercise 12.1 | Q 4.1 | Page 271

Fill in the blanks:

The x-axis and y-axis taken together determine a plane known as_______.

Exercise 12.1 | Q 4.2 | Page 271

Fill in the blanks:

The coordinates of points in the XY-plane are of the form _______.

Exercise 12.1 | Q 4.3 | Page 271

Fill in the blanks:

Coordinate planes divide the space into ______ octants.

Exercise 12.2 [Page 273]

### NCERT solutions for Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry Exercise 12.2 [Page 273]

Exercise 12.2 | Q 1.1 | Page 273

Find the distance between the pairs of points

(2, 3, 5) and (4, 3, 1)

Exercise 12.2 | Q 1.2 | Page 273

Find the distance between the pairs of points:

(–3, 7, 2) and (2, 4, –1)

Exercise 12.2 | Q 1.3 | Page 273

Find the distance between the pairs of points:

(–1, 3, –4) and (1, –3, 4)

Exercise 12.2 | Q 1.4 | Page 273

Find the distance between the following pairs of points:

(2, –1, 3) and (–2, 1, 3)

Exercise 12.2 | Q 2 | Page 273

Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.

Exercise 12.2 | Q 3.1 | Page 273

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

Exercise 12.2 | Q 3.2 | Page 273

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

Exercise 12.2 | Q 3.3 | Page 273

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

Exercise 12.2 | Q 4 | Page 273

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

Exercise 12.2 | Q 5 | Page 273

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

Exercise 12.3 [Page 277]

### NCERT solutions for Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry Exercise 12.3 [Page 277]

Exercise 12.3 | Q 1 | Page 277

Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6) in the ratio (i) 2:3 internally, (ii) 2:3 externally.

Exercise 12.3 | Q 2 | Page 277

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.

Exercise 12.3 | Q 3 | Page 277

Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8).

Exercise 12.3 | Q 4 | Page 277

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

Exercise 12.3 | Q 5 | Page 277

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

Miscellaneous Exercise [Pages 278 - 279]

### NCERT solutions for Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry Miscellaneous Exercise [Pages 278 - 279]

Miscellaneous Exercise | Q 1 | Page 278

Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) andC (–1, 1, 2). Find the coordinates of the fourth vertex.

Miscellaneous Exercise | Q 2 | Page 278

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).

Miscellaneous Exercise | Q 3 | Page 278

If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of ab and c

Miscellaneous Exercise | Q 4 | Page 279

Find the coordinates of a point on y-axis which are at a distance of 5sqrt2 from the point P (3, –2, 5).

Miscellaneous Exercise | Q 5 | Page 279

A point R with x-coordinate 4 lies on the line segment joining the pointsP (2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R.

[Hint suppose R divides PQ in the ratio k: 1. The coordinates of the point R are given by ((8k + 2)/(k+1), (-3)/(k+1), (10k + 4)/(k+1))

Miscellaneous Exercise | Q 6 | Page 279

If A and B be the points (3, 4, 5) and (–1, 3, –7), respectively, find the equation of the set of points P such that PA2 + PB2 = k2, where k is a constant.

## Chapter 12: Introduction to Three Dimensional Geometry

Exercise 12.1Exercise 12.2Exercise 12.3Miscellaneous Exercise ## NCERT solutions for Class 11 Mathematics chapter 12 - Introduction to Three Dimensional Geometry

NCERT solutions for Class 11 Mathematics chapter 12 (Introduction to Three Dimensional 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 Class 11 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 12 Introduction to Three Dimensional Geometry are Coordinate Axes and Coordinate planes, Coordinates of a Point in Space, Distance Between Two Points, Section Formula, Three Dimessional Space.

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