NCERT Textbook: Coordinate Geometry | Mathematics (Maths) Class 9 PDF Download

 Page 1


CHAPTER 3
COORDINATE GEOMETRY
What’s the good of Mercator’s North Poles and Equators, Tropics, Zones and
Meridian Lines?’ So the Bellman would cry; and crew would reply ‘ They are
merely conventional signs!’
LEWIS CARROLL, The Hunting of the Snark
3.1 Introduction
You have already studied how to locate a point on a number line. You also know how
to describe the position of a point on the line. There are many other situations, in which
to find a point we are required to describe its position with reference to more than one
line. For example, consider the following situations:
I. In Fig. 3.1, there is a main road running
in the East-West direction and streets with
numbering from West to East. Also, on each
street, house numbers are marked. To look for
a friend’s house here, is it enough to know only
one reference point? For instance, if we only
know that she lives on Street 2, will we be able
to find her house easily? Not as easily as when
we know two pieces of information about it,
namely, the number of the street on which it is
situated, and the house number. If we want to
reach the house which is situated in the 2
nd
street and has the number 5, first of all we
would identify the 2
nd
 street and then the house
numbered 5 on it. In Fig. 3.1, H shows the
location of the house. Similarly, P shows the
location of the house corresponding to Street
number 7 and House number 4.
Fig. 3.1
2024-25
Page 2


CHAPTER 3
COORDINATE GEOMETRY
What’s the good of Mercator’s North Poles and Equators, Tropics, Zones and
Meridian Lines?’ So the Bellman would cry; and crew would reply ‘ They are
merely conventional signs!’
LEWIS CARROLL, The Hunting of the Snark
3.1 Introduction
You have already studied how to locate a point on a number line. You also know how
to describe the position of a point on the line. There are many other situations, in which
to find a point we are required to describe its position with reference to more than one
line. For example, consider the following situations:
I. In Fig. 3.1, there is a main road running
in the East-West direction and streets with
numbering from West to East. Also, on each
street, house numbers are marked. To look for
a friend’s house here, is it enough to know only
one reference point? For instance, if we only
know that she lives on Street 2, will we be able
to find her house easily? Not as easily as when
we know two pieces of information about it,
namely, the number of the street on which it is
situated, and the house number. If we want to
reach the house which is situated in the 2
nd
street and has the number 5, first of all we
would identify the 2
nd
 street and then the house
numbered 5 on it. In Fig. 3.1, H shows the
location of the house. Similarly, P shows the
location of the house corresponding to Street
number 7 and House number 4.
Fig. 3.1
2024-25
44 MATHEMA TICS
II. Suppose you put a dot on a sheet of paper [Fig.3.2 (a)]. If we ask you to tell us
the position of the dot on the paper, how will you do this? Perhaps you will try in some
such manner: “The dot is in the upper half of the paper”, or “It is near the left edge of
the paper”, or “It is very near the left hand upper corner of the sheet”. Do any of
these statements fix the position of the dot precisely? No! But, if you say “ The dot is
nearly 5 cm away from the left edge of the paper”, it helps to give some idea but still
does not fix the position of the dot. A little thought might enable you to say that the dot
is also at a distance of 9 cm above the bottom line. W e now know exactly where the dot is!
Fig. 3.2
For this purpose, we fixed the position of the dot by specifying its distances from two
fixed lines, the left edge of the paper and the bottom line of the paper [Fig.3.2 (b)]. In
other words, we need two independent informations for finding the position of the dot.
Now, perform the following classroom activity known as ‘Seating Plan’.
Activity 1 (Seating Plan) : Draw a plan of the seating in your classroom, pushing all
the desks together. Represent each desk by a square. In each square, write the name
of the student occupying the desk, which the square represents. Position of each
student in the classroom is described precisely by using two independent informations:
(i) the column in which she or he sits,
(ii) the row in which she or he sits.
If you are sitting on the desk lying in the 5
th
 column and 3
rd
 row (represented by
the shaded square in Fig. 3.3), your position could be written as (5, 3), first writing the
column number, and then the row number. Is this the same as (3, 5)? Write down the
names and positions of other students in your class. For example, if Sonia is sitting in
the 4
th
 column and 1
st
 row, write S(4,1). The teacher’s desk is not part of your seating
plan. We are treating the teacher just as an observer.
2024-25
Page 3


CHAPTER 3
COORDINATE GEOMETRY
What’s the good of Mercator’s North Poles and Equators, Tropics, Zones and
Meridian Lines?’ So the Bellman would cry; and crew would reply ‘ They are
merely conventional signs!’
LEWIS CARROLL, The Hunting of the Snark
3.1 Introduction
You have already studied how to locate a point on a number line. You also know how
to describe the position of a point on the line. There are many other situations, in which
to find a point we are required to describe its position with reference to more than one
line. For example, consider the following situations:
I. In Fig. 3.1, there is a main road running
in the East-West direction and streets with
numbering from West to East. Also, on each
street, house numbers are marked. To look for
a friend’s house here, is it enough to know only
one reference point? For instance, if we only
know that she lives on Street 2, will we be able
to find her house easily? Not as easily as when
we know two pieces of information about it,
namely, the number of the street on which it is
situated, and the house number. If we want to
reach the house which is situated in the 2
nd
street and has the number 5, first of all we
would identify the 2
nd
 street and then the house
numbered 5 on it. In Fig. 3.1, H shows the
location of the house. Similarly, P shows the
location of the house corresponding to Street
number 7 and House number 4.
Fig. 3.1
2024-25
44 MATHEMA TICS
II. Suppose you put a dot on a sheet of paper [Fig.3.2 (a)]. If we ask you to tell us
the position of the dot on the paper, how will you do this? Perhaps you will try in some
such manner: “The dot is in the upper half of the paper”, or “It is near the left edge of
the paper”, or “It is very near the left hand upper corner of the sheet”. Do any of
these statements fix the position of the dot precisely? No! But, if you say “ The dot is
nearly 5 cm away from the left edge of the paper”, it helps to give some idea but still
does not fix the position of the dot. A little thought might enable you to say that the dot
is also at a distance of 9 cm above the bottom line. W e now know exactly where the dot is!
Fig. 3.2
For this purpose, we fixed the position of the dot by specifying its distances from two
fixed lines, the left edge of the paper and the bottom line of the paper [Fig.3.2 (b)]. In
other words, we need two independent informations for finding the position of the dot.
Now, perform the following classroom activity known as ‘Seating Plan’.
Activity 1 (Seating Plan) : Draw a plan of the seating in your classroom, pushing all
the desks together. Represent each desk by a square. In each square, write the name
of the student occupying the desk, which the square represents. Position of each
student in the classroom is described precisely by using two independent informations:
(i) the column in which she or he sits,
(ii) the row in which she or he sits.
If you are sitting on the desk lying in the 5
th
 column and 3
rd
 row (represented by
the shaded square in Fig. 3.3), your position could be written as (5, 3), first writing the
column number, and then the row number. Is this the same as (3, 5)? Write down the
names and positions of other students in your class. For example, if Sonia is sitting in
the 4
th
 column and 1
st
 row, write S(4,1). The teacher’s desk is not part of your seating
plan. We are treating the teacher just as an observer.
2024-25
COORDINATE GEOMETRY 45
Fig. 3.3
In the discussion above, you observe that position of any object lying in a plane
can be represented with the help of two perpendicular lines. In case of ‘dot’, we
require distance of the dot from bottom line as well as from left edge of the paper. In
case of seating plan, we require the number of the column and that of the row. This
simple idea has far reaching consequences, and has given rise to a very important
branch of Mathematics known as Coordinate Geometry. In this chapter, we aim to
introduce some basic concepts of coordinate geometry. You will study more about
these in your higher classes. This study was initially developed by the French philosopher
and mathematician René Déscartes.
René Déscartes, the great French mathematician of the
seventeenth century, liked to lie in bed and think! One
day, when resting in bed, he solved the problem of
describing the position of a point in a plane. His method
was a development of the older idea of latitude and
longitude. In honour of Déscartes, the system used for
describing the position of a point in a plane is also
known as the Cartesian system.
EXERCISE 3.1
1. How will you describe the position of a table lamp on your study table to another
person?
2. (Street Plan) : A city has two main roads which cross each other at the centre of the
city. These two roads are along the North-South direction and East-West direction.
René Déscartes (1596 -1650)
Fig. 3.4
2024-25
Page 4


CHAPTER 3
COORDINATE GEOMETRY
What’s the good of Mercator’s North Poles and Equators, Tropics, Zones and
Meridian Lines?’ So the Bellman would cry; and crew would reply ‘ They are
merely conventional signs!’
LEWIS CARROLL, The Hunting of the Snark
3.1 Introduction
You have already studied how to locate a point on a number line. You also know how
to describe the position of a point on the line. There are many other situations, in which
to find a point we are required to describe its position with reference to more than one
line. For example, consider the following situations:
I. In Fig. 3.1, there is a main road running
in the East-West direction and streets with
numbering from West to East. Also, on each
street, house numbers are marked. To look for
a friend’s house here, is it enough to know only
one reference point? For instance, if we only
know that she lives on Street 2, will we be able
to find her house easily? Not as easily as when
we know two pieces of information about it,
namely, the number of the street on which it is
situated, and the house number. If we want to
reach the house which is situated in the 2
nd
street and has the number 5, first of all we
would identify the 2
nd
 street and then the house
numbered 5 on it. In Fig. 3.1, H shows the
location of the house. Similarly, P shows the
location of the house corresponding to Street
number 7 and House number 4.
Fig. 3.1
2024-25
44 MATHEMA TICS
II. Suppose you put a dot on a sheet of paper [Fig.3.2 (a)]. If we ask you to tell us
the position of the dot on the paper, how will you do this? Perhaps you will try in some
such manner: “The dot is in the upper half of the paper”, or “It is near the left edge of
the paper”, or “It is very near the left hand upper corner of the sheet”. Do any of
these statements fix the position of the dot precisely? No! But, if you say “ The dot is
nearly 5 cm away from the left edge of the paper”, it helps to give some idea but still
does not fix the position of the dot. A little thought might enable you to say that the dot
is also at a distance of 9 cm above the bottom line. W e now know exactly where the dot is!
Fig. 3.2
For this purpose, we fixed the position of the dot by specifying its distances from two
fixed lines, the left edge of the paper and the bottom line of the paper [Fig.3.2 (b)]. In
other words, we need two independent informations for finding the position of the dot.
Now, perform the following classroom activity known as ‘Seating Plan’.
Activity 1 (Seating Plan) : Draw a plan of the seating in your classroom, pushing all
the desks together. Represent each desk by a square. In each square, write the name
of the student occupying the desk, which the square represents. Position of each
student in the classroom is described precisely by using two independent informations:
(i) the column in which she or he sits,
(ii) the row in which she or he sits.
If you are sitting on the desk lying in the 5
th
 column and 3
rd
 row (represented by
the shaded square in Fig. 3.3), your position could be written as (5, 3), first writing the
column number, and then the row number. Is this the same as (3, 5)? Write down the
names and positions of other students in your class. For example, if Sonia is sitting in
the 4
th
 column and 1
st
 row, write S(4,1). The teacher’s desk is not part of your seating
plan. We are treating the teacher just as an observer.
2024-25
COORDINATE GEOMETRY 45
Fig. 3.3
In the discussion above, you observe that position of any object lying in a plane
can be represented with the help of two perpendicular lines. In case of ‘dot’, we
require distance of the dot from bottom line as well as from left edge of the paper. In
case of seating plan, we require the number of the column and that of the row. This
simple idea has far reaching consequences, and has given rise to a very important
branch of Mathematics known as Coordinate Geometry. In this chapter, we aim to
introduce some basic concepts of coordinate geometry. You will study more about
these in your higher classes. This study was initially developed by the French philosopher
and mathematician René Déscartes.
René Déscartes, the great French mathematician of the
seventeenth century, liked to lie in bed and think! One
day, when resting in bed, he solved the problem of
describing the position of a point in a plane. His method
was a development of the older idea of latitude and
longitude. In honour of Déscartes, the system used for
describing the position of a point in a plane is also
known as the Cartesian system.
EXERCISE 3.1
1. How will you describe the position of a table lamp on your study table to another
person?
2. (Street Plan) : A city has two main roads which cross each other at the centre of the
city. These two roads are along the North-South direction and East-West direction.
René Déscartes (1596 -1650)
Fig. 3.4
2024-25
46 MATHEMA TICS
All the other streets of the city run parallel to these roads and are 200 m apart. There
are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your
notebook. Represent the roads/streets by single lines.
There are many cross- streets in your model. A particular cross-street is made by
two streets, one running in the North - South direction and another in the East - West
direction. Each cross street is referred to in the following manner : If the 2
nd
 street
running in the North - South direction and 5
th
 in the East - W est direction meet at some
crossing, then we will call this cross-street (2, 5). Using this convention, find:
(i) how many cross - streets can be referred to as (4, 3).
(ii) how many cross - streets can be referred to as (3, 4).
3.2 Cartesian System
You have studied the number line in the chapter on ‘Number System’. On the number
line, distances from a fixed point are marked in equal units positively in one direction
and negatively in the other. The point from which the distances are marked is called
the origin. We use the number line to represent the numbers by marking points on a
line at equal distances. If one unit distance represents the number ‘1’, then 3 units
distance represents the number ‘3’, ‘0’ being at the origin. The point in the positive
direction at a distance r from the origin represents the number r. The point in the
negative direction at a distance r from the origin represents the number -r. Locations
of different numbers on the number line are shown in Fig. 3.5.
Fig. 3.5
Descartes invented the idea of placing two such lines perpendicular to each other
on a plane, and locating points on the plane by referring them to these lines. The
perpendicular lines may be in any direction such as in Fig.3.6. But, when we choose
Fig. 3.6
2024-25
Page 5


CHAPTER 3
COORDINATE GEOMETRY
What’s the good of Mercator’s North Poles and Equators, Tropics, Zones and
Meridian Lines?’ So the Bellman would cry; and crew would reply ‘ They are
merely conventional signs!’
LEWIS CARROLL, The Hunting of the Snark
3.1 Introduction
You have already studied how to locate a point on a number line. You also know how
to describe the position of a point on the line. There are many other situations, in which
to find a point we are required to describe its position with reference to more than one
line. For example, consider the following situations:
I. In Fig. 3.1, there is a main road running
in the East-West direction and streets with
numbering from West to East. Also, on each
street, house numbers are marked. To look for
a friend’s house here, is it enough to know only
one reference point? For instance, if we only
know that she lives on Street 2, will we be able
to find her house easily? Not as easily as when
we know two pieces of information about it,
namely, the number of the street on which it is
situated, and the house number. If we want to
reach the house which is situated in the 2
nd
street and has the number 5, first of all we
would identify the 2
nd
 street and then the house
numbered 5 on it. In Fig. 3.1, H shows the
location of the house. Similarly, P shows the
location of the house corresponding to Street
number 7 and House number 4.
Fig. 3.1
2024-25
44 MATHEMA TICS
II. Suppose you put a dot on a sheet of paper [Fig.3.2 (a)]. If we ask you to tell us
the position of the dot on the paper, how will you do this? Perhaps you will try in some
such manner: “The dot is in the upper half of the paper”, or “It is near the left edge of
the paper”, or “It is very near the left hand upper corner of the sheet”. Do any of
these statements fix the position of the dot precisely? No! But, if you say “ The dot is
nearly 5 cm away from the left edge of the paper”, it helps to give some idea but still
does not fix the position of the dot. A little thought might enable you to say that the dot
is also at a distance of 9 cm above the bottom line. W e now know exactly where the dot is!
Fig. 3.2
For this purpose, we fixed the position of the dot by specifying its distances from two
fixed lines, the left edge of the paper and the bottom line of the paper [Fig.3.2 (b)]. In
other words, we need two independent informations for finding the position of the dot.
Now, perform the following classroom activity known as ‘Seating Plan’.
Activity 1 (Seating Plan) : Draw a plan of the seating in your classroom, pushing all
the desks together. Represent each desk by a square. In each square, write the name
of the student occupying the desk, which the square represents. Position of each
student in the classroom is described precisely by using two independent informations:
(i) the column in which she or he sits,
(ii) the row in which she or he sits.
If you are sitting on the desk lying in the 5
th
 column and 3
rd
 row (represented by
the shaded square in Fig. 3.3), your position could be written as (5, 3), first writing the
column number, and then the row number. Is this the same as (3, 5)? Write down the
names and positions of other students in your class. For example, if Sonia is sitting in
the 4
th
 column and 1
st
 row, write S(4,1). The teacher’s desk is not part of your seating
plan. We are treating the teacher just as an observer.
2024-25
COORDINATE GEOMETRY 45
Fig. 3.3
In the discussion above, you observe that position of any object lying in a plane
can be represented with the help of two perpendicular lines. In case of ‘dot’, we
require distance of the dot from bottom line as well as from left edge of the paper. In
case of seating plan, we require the number of the column and that of the row. This
simple idea has far reaching consequences, and has given rise to a very important
branch of Mathematics known as Coordinate Geometry. In this chapter, we aim to
introduce some basic concepts of coordinate geometry. You will study more about
these in your higher classes. This study was initially developed by the French philosopher
and mathematician René Déscartes.
René Déscartes, the great French mathematician of the
seventeenth century, liked to lie in bed and think! One
day, when resting in bed, he solved the problem of
describing the position of a point in a plane. His method
was a development of the older idea of latitude and
longitude. In honour of Déscartes, the system used for
describing the position of a point in a plane is also
known as the Cartesian system.
EXERCISE 3.1
1. How will you describe the position of a table lamp on your study table to another
person?
2. (Street Plan) : A city has two main roads which cross each other at the centre of the
city. These two roads are along the North-South direction and East-West direction.
René Déscartes (1596 -1650)
Fig. 3.4
2024-25
46 MATHEMA TICS
All the other streets of the city run parallel to these roads and are 200 m apart. There
are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your
notebook. Represent the roads/streets by single lines.
There are many cross- streets in your model. A particular cross-street is made by
two streets, one running in the North - South direction and another in the East - West
direction. Each cross street is referred to in the following manner : If the 2
nd
 street
running in the North - South direction and 5
th
 in the East - W est direction meet at some
crossing, then we will call this cross-street (2, 5). Using this convention, find:
(i) how many cross - streets can be referred to as (4, 3).
(ii) how many cross - streets can be referred to as (3, 4).
3.2 Cartesian System
You have studied the number line in the chapter on ‘Number System’. On the number
line, distances from a fixed point are marked in equal units positively in one direction
and negatively in the other. The point from which the distances are marked is called
the origin. We use the number line to represent the numbers by marking points on a
line at equal distances. If one unit distance represents the number ‘1’, then 3 units
distance represents the number ‘3’, ‘0’ being at the origin. The point in the positive
direction at a distance r from the origin represents the number r. The point in the
negative direction at a distance r from the origin represents the number -r. Locations
of different numbers on the number line are shown in Fig. 3.5.
Fig. 3.5
Descartes invented the idea of placing two such lines perpendicular to each other
on a plane, and locating points on the plane by referring them to these lines. The
perpendicular lines may be in any direction such as in Fig.3.6. But, when we choose
Fig. 3.6
2024-25
COORDINATE GEOMETRY 47
these two lines to locate a point in a plane in this chapter, one line
will be horizontal and the other will be vertical, as in Fig. 3.6(c).
These lines are actually obtained as follows : Take two number
lines, calling them X'X and Y'Y . Place X'X horizontal [as in Fig. 3.7(a)]
and write the numbers on it just as written on the number line. We do
the same thing with Y'Y except that Y'Y is vertical, not horizontal
[Fig. 3.7(b)].
Fig. 3.7
Combine both the lines in such
a way that the two lines cross each
other at their zeroes, or origins
(Fig. 3.8). The horizontal line X'X
is called the x - axis and the vertical
line YY' is called the y - axis. The
point where X'X and Y'Y cross is
called the origin, and is denoted
by O. Since the positive numbers
lie on the directions OX and OY,
OX and OY are called the positive
directions of the x - axis and the
y - axis, respectively . Similarly, OX'
and OY' are called the negative
directions of the x - axis and the
y - axis, respectively.
Fig. 3.8
2024-25