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 EastWest 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 202021 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 EastWest 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 202021 52 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. 202021 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 EastWest 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 202021 52 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. 202021 COORDINATE GEOMETRY 53 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 NorthSouth direction and EastWest direction. René Déscartes (1596 1650) Fig. 3.4 202021 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 EastWest 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 202021 52 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. 202021 COORDINATE GEOMETRY 53 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 NorthSouth direction and EastWest direction. René Déscartes (1596 1650) Fig. 3.4 202021 54 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 crossstreet 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 crossstreet (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 202021 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 EastWest 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 202021 52 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. 202021 COORDINATE GEOMETRY 53 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 NorthSouth direction and EastWest direction. René Déscartes (1596 1650) Fig. 3.4 202021 54 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 crossstreet 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 crossstreet (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 202021 COORDINATE GEOMETRY 55 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 202021Read More
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