Coordinate Geometry - NCERT Solutions

Exercise 3.1
Exercise 3.2
Examples
Summary

Exercise 3.1

1. How will you describe the position of a table lamp on your study table to another person?

To describe the position of a table lamp on a study table, we can use a coordinate system. We can take two perpendicular edges of the table as reference lines (like x-axis and y-axis). Then we can specify the distance of the lamp from each edge. For example, we can say "The lamp is 30 cm from the left edge and 40 cm from the bottom edge."

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. 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 2nd street running in the North-South direction and 5th in the East-West 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).

Answer: Only one cross-street can be referred to as (4, 3).

(ii) how many cross-streets can be referred to as (3, 4).

Answer: Only one cross-street can be referred to as (3, 4).

Exercise 3.2

1. Write the answer of each of the following questions:

(i) What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?

Answer: The horizontal line is called the x-axis and the vertical line is called the y-axis.

(ii) What is the name of each part of the plane formed by these two lines?

Answer: Each part is called a quadrant.

(iii) Write the name of the point where these two lines intersect.

Answer: The point where these two lines intersect is called the origin.

2. See Fig.3.14, and write the following:

(i) The coordinates of B.

Answer: (-5, 2)

(ii) The coordinates of C.

Answer: (5, -5)

(iii) The point identified by the coordinates (-3, -5).

Answer: E

(iv) The point identified by the coordinates (2, -4).

Answer: G

(v) The abscissa of the point D.

Answer: 6

(vi) The ordinate of the point H.

Answer: -3

(vii) The coordinates of the point L.

Answer: (0, 5)

(viii) The coordinates of the point M.

Answer: (-3, 0)

Important Examples

Example 1: See Fig. 3.11 and complete the following statements:

(i) The abscissa and the ordinate of the point B are 4 and 3, respectively. Hence, the coordinates of B are (4, 3).

(ii) The x-coordinate and the y-coordinate of the point M are -3 and 4, respectively. Hence, the coordinates of M are (-3, 4).

(iii) The x-coordinate and the y-coordinate of the point L are -5 and -4, respectively. Hence, the coordinates of L are (-5, -4).

(iv) The x-coordinate and the y-coordinate of the point S are 3 and -4, respectively. Hence, the coordinates of S are (3, -4).

Example 2: Write the coordinates of the points marked on the axes in Fig. 3.12.

(i) The coordinates of A are (4, 0)

(ii) The coordinates of B are (0, 3)

(iii) The coordinates of C are (-5, 0)

(iv) The coordinates of D are (0, -4)

(v) The coordinates of E are (2/3, 0)

Key Concepts:

  • Cartesian System: A system for describing the position of a point in a plane using two perpendicular lines called axes.
  • Origin: The point of intersection of the x-axis and y-axis, denoted by O(0, 0).
  • Quadrants: The four parts into which the coordinate axes divide the plane.
  • Coordinates: An ordered pair (x, y) that describes the position of a point, where x is the distance from the y-axis (abscissa) and y is the distance from the x-axis (ordinate).

Chapter Summary

In this chapter, you have studied the following points:

  1. To locate the position of an object or a point in a plane, we require two perpendicular lines. One of them is horizontal, and the other is vertical.
  2. The plane is called the Cartesian, or coordinate plane and the lines are called the coordinate axes.
  3. The horizontal line is called the x-axis, and the vertical line is called the y-axis.
  4. The coordinate axes divide the plane into four parts called quadrants.
  5. The point of intersection of the axes is called the origin.
  6. The distance of a point from the y-axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.
  7. If the abscissa of a point is x and the ordinate is y, then (x, y) are called the coordinates of the point.
  8. The coordinates of a point on the x-axis are of the form (x, 0) and that of the point on the y-axis are (0, y).
  9. The coordinates of the origin are (0, 0).
  10. The coordinates of a point are of the form (+, +) in the first quadrant, (-, +) in the second quadrant, (-, -) in the third quadrant and (+, -) in the fourth quadrant, where + denotes a positive real number and - denotes a negative real number.
  11. If x ≠ y, then (x, y) ≠ (y, x), and (x, y) = (y, x), if x = y.

Historical Note:

René Descartes (1596-1650), the great French mathematician of the seventeenth century, developed the Cartesian coordinate system. His method was a development of the older idea of latitude and longitude. In honour of Descartes, the system used for describing the position of a point in a plane is also known as the Cartesian system.