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# NCERT solutions Mathematics Class 11 chapter 12 Introduction to Three Dimensional Geometry

## Chapter 12 - Introduction to Three Dimensional Geometry

#### Page 271

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

Q 1 | Page 271

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

Q 2 | 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)

Q 3 | Page 271

Fill in the blanks:

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

Q 4.1 | Page 271

Fill in the blanks:

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

Q 4.2 | Page 271

Fill in the blanks:

Coordinate planes divide the space into ______ octants.

Q 4.3 | Page 271

#### Page 273

Find the distance between the pairs of points

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

Q 1.1 | Page 273

Find the distance between the pairs of points:

(–3, 7, 2) and (2, 4, –1)

Q 1.2 | Page 273

Find the distance between the pairs of points:

(–1, 3, –4) and (1, –3, 4)

Q 1.3 | Page 273

Find the distance between the following pairs of points:

(2, –1, 3) and (–2, 1, 3)

Q 1.4 | Page 273

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

Q 2 | Page 273

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

Q 3.1 | Page 273

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

Q 3.2 | Page 273

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

Q 3.3 | Page 273

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

Q 4 | 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.

Q 5 | Page 273

#### 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.

Q 1 | 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.

Q 2 | 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).

Q 3 | 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.

Q 4 | 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).

Q 5 | Page 277

#### 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.

Q 1 | 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).

Q 2 | 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

Q 3 | Page 278

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

Q 4 | 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))

Q 5 | 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.

Q 6 | Page 279

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