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#### Chapters

1 - Sets2 - Relations and Functions

3 - Trigonometric Functions

4 - Principle of Mathematical Induction

5 - Complex Numbers and Quadratic Equations

6 - Linear Inequalities

7 - Permutations and Combinations

8 - Binomial Theorem

9 - Sequences and Series

10 - Straight Lines

11 - Conic Sections

12 - Introduction to Three Dimensional Geometry

13 - Limits and Derivatives

14 - Mathematical Reasoning

15 - Statistics

16 - Probability

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### Chapter 12 - Introduction to Three Dimensional Geometry

#### Page 271

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

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

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)

Fill in the blanks:

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

Fill in the blanks:

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

Fill in the blanks:

Coordinate planes divide the space into ______ octants.

#### Page 273

Find the distance between the pairs of points

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

Find the distance between the pairs of points:

(–3, 7, 2) and (2, 4, –1)

Find the distance between the pairs of points:

(–1, 3, –4) and (1, –3, 4)

Find the distance between the following pairs of points:

(2, –1, 3) and (–2, 1, 3)

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

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

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

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

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

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.

#### 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.

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.

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

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

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

#### Pages 278 - 279

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.

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

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

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

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))`

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 PA^{2} + PB^{2} = *k*^{2}, where *k* is a constant.

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