Chapter 3. Coordinate Geometry

Important Note :

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  -axis, and the vertical line is called the  - 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  - axis is called its -coordinate, or abscissa, and the distance of the point from the -axis is called its -coordinate, or ordinate.
7. If the abscissa of a point is  and the ordinate is , then () are called the coordinates of the point.
8. The coordinates of a point on the -axis are of the form () and that of the point on the -axis are ().
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 , then  and  if  .

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EXERCISE 3.1

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

Solution:  Consider the lamp as a point and table as a plane . Choose any two perpendicular edges of the table . Measure the distance of the lamp from the longer edge, suppose it is 15 cm . Again, measure the distance of the lamp from the shorter edge , and suppose it is 25 cm . We can write the position of the lamps as (15,25) , depending on the order you fix .

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 streets is referred to the following manner : If the  street running in the North –South direction and  in the East-West direction meet at some crossing , then we will call this cross-street  . Using this convention , find :  (i) how many cross-streets can be referred to as    (ii) how many cross-streets can be referred to as  .

Solution: The street plan is shown in figure given below .

Both the cross-streets are marked street plan figure .   

(i) The street 4 and street 3 are both the cross-streets can be referred to as  .

(ii) The street 3 and street 4 are both the cross-streets can be referred to as  .

The two cross-streets are uniquely found because of the two reference lines we have used for locating them .

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 ?

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

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

Solution:  (i) the name of horizontal is the x-axis and the vertical lines is the y-axis .

(ii) the name of each part of the plane formed by these two lines is quadrants .

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

2. See Fig. 3.14 and write the following :

 (i) The coordinates of B .     

(ii) The coordinates of C . 

(iii) The point identified by the coordinates  .

(iv) The point identified by the coordinates  .

(v) The abscissa of the point D .

(vi) The ordinate of the point H .

(vii) The coordinates of the point L .

(viii) The coordinates of the point M .

Solution: (i) The coordinates of B is ( – 5 , 2) .     

(ii) The coordinates of C is (5 , – 5) 

(iii) The point identified by the coordinates  is E .

(iv) The point identified by the coordinates  is G .

(v) The abscissa of the point D is 6 .

(vi) The ordinate of the point H is – 3 .

(vii) The coordinates of the point L is (0 , 5) .

(viii) The coordinates of the point M is (– 3 ,0) .

EXERCISE  3.3

1. In which quadrant or on which axis do each of the points , , and  lie ? Verify your answer by locating them on the Cartesian plane .

Solution: Taking 1square = 1unit , we draw the x-axis and the y-axis . The positions of the points are show by dots in given figure .

The point (-2,4) lies in quadrant II ,

The point (3,-1) lie in quadrant IV ,

The point (-1,0) lie on the negative x-axis ,

The point (1,2) lie in quadrant I and

The point (-3,-5) lie in quadrant III .

2. Plot the points given in the following table on the plane , choosing suitable units of the distance on the axis .

Solution: The pairs of numbers given in the table can be represented by the points (-2 , 8) ,(-1 , 7) , (0 , - 1.25) , (1 , 3) and (3 , - 1) . The locations of the points are shown by dots in given figure .

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