NCERT Solutions
Class 11 - Mathematics

Chapter 11: INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

A point is on the x-axis. What are its y-coordinate 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:

(i) The x-axis and y-axis taken together determine a plane known as ______.

(ii) The coordinates of points in the XY-plane are of the form ______.

(iii) Coordinate planes divide the space into _____ octants.

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 \((-2, 1, 3)\)

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

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.

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.

Three vertices of a parallelogram ABCD are A(3, −1, 2), B(1, 2, −4) and C(−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 C(6, 0, 0).

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), find the values of \(a\), \(b\) and \(c\).

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.