# NCERT Solutions for Class 11 Maths Maths Chapter 12

## Introduction to Three Dimensional Geometry Class 11

### Exercise 12.1 : Solutions of Questions on Page Number : 271

Q1 :

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

If a point is on the x-axis, then its y-coordinates and z-coordinates are zero.

Q2 :

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

If a point is in the XZ plane, then its y-coordinate is zero.

Q3 :

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)

Q4 :

Fill in the blanks:

### Exercise 12.2 : Solutions of Questions on Page Number : 273

Q1 :

Find the distance between the following pairs of points:

(i) (2, 3, 5) and (4, 3, 1) (ii) (-3, 7, 2) and (2, 4, -1)

(iii) (-1, 3, -4) and (1, -3, 4) (iv) (2, -1, 3) and (-2, 1, 3)

Q2 :

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

Q3 :

Verify the following:

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

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

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

Q4 :

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

Q5 :

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 : Solutions of Questions on Page Number : 277

Q1 :

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.

Q2 :

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.

Q3 :

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

Q4 :

Using section formula, show that the points A (2, â€“3, 4), B (â€“1, 2, 1) and are collinear.

Q5 :

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

### Exercise Miscellaneous : Solutions of Questions on Page Number : 278

Q1 :

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.

Q2 :

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

Q3 :

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 a, b and c.

Q4 :

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

Q5 :

A point R with x-coordinate 4 lies on the line segment joining the pointsP (2, â€“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 Q6 :

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.