Introduction to Three Dimensional Geometry

If A and B are the points (-2,2,3) and(-1,4,-3) respectively, then find the
locus of P such that 3|PA| = 2|PB|

If A and B be the points (3,4,5) and(-1,3,7) respectively. Find the eq. of the

set points P such that PA² + PB² = K² where K is a constant

Three consecutive vertices of a parallelogram ABCD are A(3,-1,2), B(1,2,-4)
and C(-1,1,2) find forth vertex D

If the points P(1,0,-6), Q(-3,P,q) and R(-5,9,6) are collinear, find the
values of P and q

A point R with x co-ordinates 4 lies on the line segment joining the points
P(2,-3,4)and Q(8,0,10)find the co-ordinates of the point R

Show that the point P(1,2,3), Q(-1,-2,-1), R(2,3, 2) and S (4,7,6)taken in
order form the vertices of a parallelogram. Do these form a rectangle?

Find the co-ordinates of the points which trisects the line segment PQ formed
by joining the point P(4,2,-6) and Q(10,-16,6)

The distance from origin to point (a,b,c) is:

The point (-4, 2,5) lie in which octants.

What is the coordinator of XY plane

A point is in the XZ plane. What is the value of y co-ordinates?

The mid points of the sides of a ΔABC are given by
(-2,3,5), (4,-1,7) and (6,5,3)find the co ordinate of A, B and C

Find the ratio in which the join the A(2,1,5)and B(3, 4,3) is divided by the
plane 2x + 2y - 2z =1Also find the co ordinate of the point of division

Find the co ordinate of a point equidistant from the four points
0(0,0,0), A(a,0,0), B(0,b,0) and C(0,0,c)

Find the co ordinate of the point which divider the join of P(2,-1, 4) and
Q(4,3,2) in the ratio 2 : 5 (i) internally (ii) externally

Prove by distance formula that the points A(1,2,3), B(-1,-1,-1) and
C(3,5,7) are collinear.

The point (2,0,8) lie in which plane

The point (4,-3,-6) lie in which octants

Write the name of plane in which x axis and y - axis taken together.

A point lie on X –axis what are co ordinate of the point

The mid points of the sides of a triangle are (1,5,-1), (0,4,-2) and (2,3,4). Find
its vertices.

Prove that the lines joining the vertices of a tetrahedron to the centroids of
the opposite faces are concurrent.

Using section formula, prove that the three points A(-2,3,5), B(1,2,3), and
C(7,0,-1) are collinear.

Show that the points A(0,1,2) B(2,-1,3) and C(1,-3,1) are vertices of an
isosceles right angled triangle.

Find the locus of the point which is equidistant from the point A(0,2,3) and
B(2,-2,1)

Determine the points in xy plane which is equidistant from these point A
(2,0,3) B(0,3,2) and C(0,0,1)

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 image of (5,2,-7) in the xy plane

Find the image of (-2,3,4) in the y z plane

Name the octants in which the following lie. (-5,4,3)

Name the octants in which the following lie. (5,2,3)

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.

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

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

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.

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

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

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

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

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 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.

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.

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

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.

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

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)

Fill in the blanks:

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)

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

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