RD Sharma solutions for Mathematics [English] Class 11 chapter 28 - Introduction to three dimensional coordinate geometry [Latest edition]

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

Name the octants in which the following points lie:

(–5, 4, 3) 

Name the octants in which the following points lie: 

(4, –3, 5)

Name the octants in which the following points lie: 

 (7, 4, –3)

Name the octants in which the following points lie: 

(–5, –4, 7) 

Name the octants in which the following points lie: 

(–5, –3, –2) 

Name the octants in which the following points lie:

 (2, –5, –7) 

Name the octants in which the following points lie: 

(–7, 2 – 5)

Find the image  of: 

 (–2, 3, 4) in the yz-plane.

Find the image  of: 

 (–5, 4, –3) in the xz-plane. 

Find the image  of:

 (5, 2, –7) in the xy-plane.

Find the image  of: 

 (–5, 0, 3) in the xz-plane. 

Find the image  of: 

 (–4, 0, 0) in the xy-plane. 

A cube of side 5 has one vertex at the point (1, 0, –1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.

Planes are drawn parallel to the coordinate planes through the points (3, 0, –1) and (–2, 5, 4). Find the lengths of the edges of the parallelepiped so formed.

Planes are drawn through the points (5, 0, 2) and (3, –2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed. 

Find the distances of the point P(–4, 3, 5) from the coordinate axes. 

The coordinates of a point are (3, –2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.

Find the distance between the following pairs of points: 

P(1, –1, 0) and Q(2, 1, 2)

Find the distance between the following pairs of point: 

A(3, 2, –1) and B(–1, –1, –1).

Find the distance between the points P and Q having coordinates (–2, 3, 1) and (2, 1, 2).

Verify the following: 

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

Using distance formula prove that the following points are collinear:

A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)

Using distance formula prove that the following points are collinear: 

P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)

Using distance formula prove that the following points are collinear: 

A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)

Determine the points in xy-plan are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).

Determine the points in yz-plane and are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).

Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1). 

Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).

Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).

Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3). 

Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.

Show that the points (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of an isosceles right-angled triangle. 

Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of a square.

Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.

Show that the points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) are the vertices of a rhombus. 

Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.

Show that the points (3, 2, 2), (–1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.

Find the coordinates of the point which is equidistant  from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).

If A(–2, 2, 3) and B(13, –3, 13) are two points.
Find the locus of a point P which moves in such a way the 3PA = 2PB.

Find the locus of P if PA² + PB² = 2k², where A and B are the points (3, 4, 5) and (–1, 3, –7).

Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle. 

Are the points A(3, 6, 9), B(10, 20, 30) and C(25, –41, 5), the vertices of a right-angled triangle?

Verify the following: 

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

Verify the following: 

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

Verify the following: 

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

Verify the following:

 (5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.

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

Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.

Show that the points A(1, 2, 3), B(–1, –2, –1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram ABCD, but not a rectangle.

Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.

The vertices of the triangle are A(5, 4, 6), B(1, –1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the length AD.

A point C with z-coordinate 8 lies on the line segment joining the points A(2, –3, 4) and B(8, 0, 10). Find its coordinates.

Show that the three points A(2, 3, 4), B(–1, 2 – 3) and C(–4, 1, –10) are collinear and find the ratio in which C divides AB. 

Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane.

Find the ratio in which the line segment joining the points (2, –1, 3) and (–1, 2, 1) is divided by the plane x + y + z = 5. 

If the points A(3, 2, –4), B(9, 8, –10) and C(5, 4, –6) are collinear, find the ratio in which Cdivides AB.

The mid-points of the sides of a triangle ABC are given by (–2, 3, 5), (4, –1, 7) and (6, 5, 3). Find the coordinates of A, B and C.

A(1, 2, 3), B(0, 4, 1), C(–1, –1, –3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle ∠BAC meets BC.

Find the ratio in which the sphere x² + y² + z² = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).

Show that the plane ax + by + cz + d = 0 divides the line joining the points (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]

Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4). 

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, –6) respectively, find the coordinates of the point C.

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

Given that  P(3, 2, –4), Q(5, 4, –6) and R(9, 8, –10) are collinear. Find the ratio in which Qdivides PR. 

Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, –8) is divided by the yz-plane. 

Write the distance of the point P (2, 3,5) from the xy-plane.

Write the distance of the point P(3, 4, 5) from z-axis.

If the distance between the points P(a, 2, 1) and Q (1, −1, 1) is 5 units, find the value of a. 

The coordinates of the mid-points of sides AB, BC and CA of  △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.

Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.

Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.

Write the coordinates of third vertex of a triangle having centroid at the origin and two vertices at (3, −5, 7) and (3, 0, 1). 

What is the locus of a point for which y = 0, z = 0?

Find the ratio in which the line segment joining the points (2, 4,5) and (3, −5, 4) is divided by the yz-plane.

Find the point on y-axis which is at a distance of  \[\sqrt{10}\] units from the point (1, 2, 3).

Find the point on x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).

Find the coordinates of a point equidistant from the origin and points A (a, 0, 0), B (0, b, 0) andC(0, 0, c). 

Write the coordinates of the point P which is five-sixth of the way from A(−2, 0, 6) to B(10, −6, −12). 

If a parallelopiped is formed by the planes drawn through the points (2,3,5) and (5, 9, 7) parallel to the coordinate planes, then write the lengths of edges of the parallelopiped and length of the diagonal. 

Determine the point on yz-plane which is equidistant from points A(2, 0, 3), B(0, 3,2) and C(0, 0,1).

If the origin is the centroid of a triangle ABC having vertices A(a, 1, 3), B(−2, b −5) and C (4, 7, c), find the values of a, b, c.

The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is

The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is

If P(0, 1, 2), Q(4, –2, 1) and O(0, 0, 0) are three points, then ∠POQ = 

If the extremities of the diagonal of a square are (1, –2, 3 and (2, –3, 5), then the length of the side is 

The points (5, –4, 2), (4, –3, 1), (7, 6, 4) and (8, –7, 5) are the vertices of 

In a three dimensional space the equation x² – 5x + 6 = 0 represents

Let (3, 4, –1) and (–1, 2, 3) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to 

XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio

What is the locus of a point for which y = 0, z = 0?

The coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are

The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are 

The perpendicular distance of the point P (6, 7, 8) from xy - plane is

The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is 

The perpendicular distance of the point P(3, 3,4) from the x-axis is 

The length of the perpendicular drawn from the point P(a, b, c) from z-axis is