#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 28: Introduction to three dimensional coordinate geometry

#### Chapter 28: Introduction to three dimensional coordinate geometry Exercise 28.10 solutions [Pages 6 - 7]

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.

#### Chapter 28: Introduction to three dimensional coordinate geometry Exercise 28.20 solutions [Pages 9 - 10]

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 3*PA** *= 2*PB*.

Find the locus of *P* if *PA*^{2} + *PB*^{2}^{ }= 2*k*^{2}, 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.

#### Chapter 28: Introduction to three dimensional coordinate geometry Exercise 28.30 solutions [Pages 19 - 20]

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 *C*divides *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 *x*^{2} + *y*^{2}^{ }+ *z*^{2} = 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 (*x*_{1}, *y*_{1}, *z*_{1}) and (*x*_{2}, *y*_{2}, *z*_{2}) 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).

Using section formula, show that he points *A*(2, –3, 4), *B*(–1, 2, 1) and *C*(0, 1/3, 2) are collinear.

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

#### Chapter 28: Introduction to three dimensional coordinate geometry solutions [Page 22]

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.*

#### Chapter 28: Introduction to three dimensional coordinate geometry solutions [Pages 22 - 23]

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

2 : 3

3 : 2

–2 : 3

4 : –3

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

*a*:*b**b*:*c**c*:*a**c*:*b*

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

\[\frac{\pi}{6}\]

\[\frac{\pi}{4}\]

\[\frac{\pi}{3}\]

\[\frac{\pi}{2}\]

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

\[\sqrt{6}\]

\[\sqrt{3}\]

\[\sqrt{5}\]

\[\sqrt{7}\]

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

a rectangle

a square

a parallelogram

none of these

In a three dimensional space the equation *x*^{2} – 5*x* + 6 = 0 represents

points

planes

curves

pair of straight lines

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

2

3

6

7

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

3 : 7

2 : 7

–3 : 7

–2 : 7

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

*x*- axis*y*- axis*z*- axis*yz*- plane

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

(3, 4, 0)

(0, 4, 5)

(3, 0, 5)

(3, 0, 0)

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

(6, 0, 0)

(0, 7, 0)

(0, 0, 8)

(0, 7, 8)

We know that the

*y*and*z*coordinates on*x*- axis are 0

The coordinates of the foot of the perpendicular from a point P(6,7, 8) on*x*- axis are (6, 0, 0)

Hence, the correct answer is option (a).

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

8

7

6

10

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

10

\[\sqrt{34}\]

\[\sqrt{113}\]

512

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

\[3\sqrt{2}\]

5

3

4

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

\[\sqrt{a^2 + b^2}\]

\[\sqrt{b^2 + c^2}\]

\[\sqrt{a^2 + c^2}\]

\[\sqrt{a^2 + b^2 + c^2}\]

## Chapter 28: Introduction to three dimensional coordinate geometry

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 28 - Introduction to three dimensional coordinate geometry

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Concepts covered in Class 11 Mathematics chapter 28 Introduction to three dimensional coordinate geometry are Coordinate Axes and Coordinate planes, Coordinates of a Point in Space, Distance Between Two Points, Section Formula, Three Dimessional Space.

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