Chapter 31.1 - Finding Coordinates - Solved examples

In the previous section we saw the procedure for locating points in the cartesian plane. In this section we will see a few solved examples.

Solved example 31.1
Observe the fig.31.8 below and complete the following statements:

Fig.31.8

(a) The abscissa and ordinate of the point B are ___ and ___ respectively. Hence, the coordinates of B are (___ , ___)
(b) The x-coordinate and y-coordinate of the point M are  ___ and ___ respectively. Hence, the coordinates of M are (___ , ___)    
(c) The x-coordinate and y-coordinate of the point L are  ___ and ___ respectively. Hence, the coordinates of L are (___ , ___)
(d) The x-coordinate and y-coordinate of the point S are  ___ and ___ respectively. Hence, the coordinates of S are (___ , ___)
Solution:
(a) Let us apply the rules to point B:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of B from the y-axis (measured along the x-axis) is '4 units'
    ♦ It is measured along OX. So it is positive
    ♦ Thus the x-coordinate of B is '4'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of B from the x-axis (measured along the y-axis) is '3 units'
    ♦ It is measured along OY. So it is positive
    ♦ Thus the y-coordinate of B is '3'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of B are: (4,3)

(b) Let us apply the rules to point M:
(i) Applying the first rule to find the x-coordinate:
• Perpendicular distance of M from the y-axis (measured along the x-axis) is '3 units'
    ♦ It is measured along OX'. So it is negative
    ♦ Thus the x-coordinate of M is '-3'
(ii) Applying the second rule to find the y-coordinate:
• Perpendicular distance of M from the x-axis (measured along the y-axis) is '4 units'
    ♦ It is measured along OY. So it is positive
    ♦ Thus the y-coordinate of M is '4'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of M are: (-3,4)

(c) Let us apply the rules to point L:
(i) Applying the first rule to find the x-coordinate:
• Perpendicular distance of L from the y-axis (measured along the x-axis) is '5 units'
    ♦ It is measured along OX'. So it is negative
    ♦ Thus the x-coordinate of L is '-5'
(ii) Applying the second rule to find the y-coordinate:
• Perpendicular distance of L from the x-axis (measured along the y-axis) is '4 units'
    ♦ It is measured along OY'. So it is negative
    ♦ Thus the y-coordinate of L is '-4'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of L are: (-5,-4)

(d) Let us apply the rules to point S:
(i) Applying the first rule to find the x-coordinate:
• Perpendicular distance of S from the y-axis (measured along the x-axis) is '3 units'
    ♦ It is measured along OX. So it is positive.
    ♦ Thus the x-coordinate of S is '3'
(ii) Applying the second rule to find the y-coordinate:
• Perpendicular distance of S from the x-axis (measured along the y-axis) is '4 units'
    ♦ It is measured along OY'. So it is negative
    ♦ Thus the y-coordinate of S is '-4'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of S are: (3,-4)

Solved example 31.2
In the fig.31.9 below, some points are marked on the axes itself. Write the coordinates of each of them. Given that, point E is at a distance of ²⁄₃ units from the origin.

Fig.31.9

Solution:
(a) Let us apply the rules to point A:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of A from the y-axis (measured along the x-axis) is '4 units'
    ♦ It is measured along OX. So it is positive
    ♦ Thus the x-coordinate of A is '4'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of A from the x-axis is '0 units'
    ♦ We cannot say that is measured along any particular direction. Because there is no measurement to take. '0 units' is neither positive nor negative
    ♦ Thus the y-coordinate of A is '0'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of A are: (4,0)

(b) Let us apply the rules to point B:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of B from the y-axis is '0 units'
    ♦ We cannot say that is measured along any particular direction. Because there is no measurement to take. '0 units' is neither positive nor negative
    ♦ Thus the x-coordinate of B is '0'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of B from the x-axis (measured along the y-axis) is '3 units'
    ♦ It is measured along OY. So it is positive
    ♦ Thus the y-coordinate of B is '3'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of B are: (0,3)

(c) Let us apply the rules to point C:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of C from the y-axis (measured along the x-axis) is '5 units'
    ♦ It is measured along OX'. So it is negative
    ♦ Thus the x-coordinate of C is '-5'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of C from the x-axis is '0 units'
    ♦ We cannot say that is measured along any particular direction. Because there is no measurement to take. '0 units' is neither positive nor negative
    ♦ Thus the y-coordinate of C is '0'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of C are: (-5,0)

(d) Let us apply the rules to point D:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of D from the y-axis is '0 units'
    ♦ We cannot say that is measured along any particular direction. Because there is no measurement to take. '0 units' is neither positive nor negative
    ♦ Thus the x-coordinate of D is '0'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of D from the x-axis (measured along the y-axis) is '4 units'
    ♦ It is measured along OY'. So it is negative
    ♦ Thus the y-coordinate of D is '-4'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of D are: (0,-4)

(e) Let us apply the rules to point E:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of E from the y-axis (measured along the x-axis) is '²⁄₃ units'
    ♦ It is measured along OX. So it is positive
    ♦ Thus the x-coordinate of E is '²⁄₃'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of E from the x-axis is '0 units'
    ♦ We cannot say that is measured along any particular direction. Because there is no measurement to take. '0 units' is neither positive nor negative
    ♦ Thus the y-coordinate of E is '0'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of E are: (²⁄₃,0)

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Based on the results from the above solved example 31.2, we can write the following:

• If a point is on the x-axis, it's y-coordinate will be zero. That is:
■ The coordinates of all points on the x-axis will be of the form: (±x,0)
    ♦ Use '+' sign if the point lies on OX
    ♦ Use '-' sign if the point lies on OX'
• If a point is on the y-axis, it's x-coordinate will be zero. That is:
■ The coordinates of all points on the y-axis will be of the form: (0,±y)
    ♦ Use '+' sign if the point lies on OY
    ♦ Use '-' sign if the point lies on OY'
• Now we can consider the origin
    ♦ The distance of O from the x-axis is 0
    ♦ The distance of O from the y-axis is 0 
■ So the coordinates of the origin will always be: (0,0)

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From the above two solved examples, we get the following information:
1. If a point lies in the first quadrant, 
• it's x-coordinate will be positive
• it's y-coordinate will be positive
• It is enclosed between the positive OX and positive OY
■ So it's coordinates will be of the form (+,+)
2. If a point lies in the second quadrant, 
• it's x-coordinate will be negative
• it's y-coordinate will be positive
• It is enclosed between the negative OX' and positive OY
■ So it's coordinates will be of the form (-,+)
3. If a point lies in the third quadrant, 
• it's x-coordinate will be negative
• it's y-coordinate will be negative
• It is enclosed between the negative OX' and negative OY'
■ So it's coordinates will be of the form (-,-)
4. If a point lies in the fourth quadrant, 
• it's x-coordinate will be positive
• it's y-coordinate will be negative
• It is enclosed between the positive OX and positive OY'
■ So it's coordinates will be of the form (+,-)
This is shown in the fig.31.10 below:

Fig.31.10

Solved example 31.3
Observe the fig.31.11 below and write the following:
1. The coordinates of B
2. The coordinates of C
3. The point identified by the coordinates (-3,-5)
4. The point identified by the coordinates (2,-4)
5. The abscissa of point D
6. The ordinate of point H
7. The coordinates of point H
8. The coordinates of point M

Solution:
(1) Let us apply the rules to point B:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of B from the y-axis (measured along the x-axis) is '5 units'
    ♦ It is measured along OX'. So it is negative
    ♦ Thus the x-coordinate of B is '-5'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of B from the x-axis (measured along the y-axis) is '2 units'
    ♦ It is measured along OY. So it is positive
    ♦ Thus the y-coordinate of B is '2'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of B are: (-5,2)

(2) Let us apply the rules to point C:
(i) Applying the first rule to find the x-coordinate:
• Perpendicular distance of C from the y-axis (measured along the x-axis) is '5 units'
    ♦ It is measured along OX. So it is positive
    ♦ Thus the x-coordinate of C is '5'
(ii) Applying the second rule to find the y-coordinate:
• Perpendicular distance of C from the x-axis (measured along the y-axis) is '5 units'
    ♦ It is measured along OY'. So it is negative
    ♦ Thus the y-coordinate of B is '-5'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of C are: (-5,-5)


(3) The given coordinates are (-3,-5)
• Here both the x and y coordinates are negative. So the point will be in the third quadrant
• So we need to consider H and E only. The required answer will be one among them
• The given x coordinate is '-3'. So the required point will be at a distance of 3 units from the y-axis
    ♦ H is at a distance of 5 units from the y axis
    ♦ E is at a distance of 3 units from the y axis
• So the required point must be E
• To confirm, let us check the given y coordinate. It is '-5'
• E is indeed at a distance of 5 units from the x-axis.
• So E is our required point

(4) The given coordinates are (2,-4)
• Here the x coordinate is positive and y coordinate is negative. So the point will be in the fourth quadrant
• So we need to consider G and C only. The required answer will be one among them
• The given x coordinate is '2'. So the required point will be at a distance of 2 units from the y-axis
    ♦ G is at a distance of 2 units from the y axis
    ♦ C is at a distance of 5 units from the y axis
• So the required point must be G
• To confirm, let us check the given y coordinate. It is '-4'
• G is indeed at a distance of 4 units from the x-axis.
• So G is our required point

(5) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of D from the y-axis (measured along the x-axis) is '6 units'
    ♦ It is measured along OX. So it is positive
    ♦ Thus the x-coordinate (abscissa) of D is '6'

(6) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of H from the x-axis (measured along the y-axis) is '3 units'
    ♦ It is measured along OY'. So it is negative
    ♦ Thus the y-coordinate (ordinate) of B is '-3'

(7) Let us apply the rules to point L:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of L from the y-axis is '0 units'
    ♦ We cannot say that is measured along any particular direction. Because there is no measurement to take. '0 units' is neither positive nor negative
    ♦ Thus the x-coordinate of L is '0'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of L from the x-axis (measured along the y-axis) is '5 units'
    ♦ It is measured along OY. So it is positive
    ♦ Thus the y-coordinate of L is '5'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of L are: (0,5)

(8) Let us apply the rules to point M:
(i) Applying the first rule to find the x-coordinate (abscissa):
• Perpendicular distance of M from the y-axis (measured along the x-axis) is '3 units'
    ♦ It is measured along OX'. So it is negative
    ♦ Thus the x-coordinate of M is '-3'
(ii) Applying the second rule to find the y-coordinate (ordinate):
• Perpendicular distance of M from the x-axis is '0 units'
    ♦ We cannot say that is measured along any particular direction. Because there is no measurement to take. '0 units' is neither positive nor negative
    ♦ Thus the y-coordinate of M is '0'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of M are: (-3,0)

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In the next section we will see the method of plotting.

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Chapter 31 - Coordinate Geometry

In the previous section we completed a discussion on Trigonometry. In this section we will see Coordinate Geometry. We saw some basic details about graphs in chapter 2. Here we will have a more detailed study.

We have studied about the number lines also in an earlier chapter 22.  Following are the basic features of a number line:
• The distances are marked from a fixed point
    ♦ This fixed point is called the origin
• The distances are marked in equal units
    ♦ If 1 unit from the origin represents the number 1, then 3 units from origin will represent the number 3
    ♦ In the same way, the number 'r' will be 'r units' away from the origin 
• The distances are marked positively in one direction and negatively in the opposite direction
    ♦ The number 'r' will be 'r units' away from the origin in the positive direction
    ♦ The number '-r' will be 'r units' away from the origin in the negative direction
• An example is shown in the fig.31.1 below:

Fig.31.1

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• Consider two such number lines. Place them perpendicular to each other. 

    ♦ But there are infinite possibilities to place 'two lines perpendicular to each other'. The xxx fig.31.2 below shows some of them:

Fig.31.2

• All the three arrangements in the above fig.31.2 satisfies one condition:

    ♦ The lines should be perpendicular to each other. 
• In this chapter, we will be using the arrangement in fig.c. That is:
    ♦ One line will be horizontal and the other line will be vertical

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Let us see how the numbering system can be applied to the two perpendicular lines:

1. Consider any simple horizontal number line as shown in the fig.31.3(a) below. Let us name it X'X
• The positive numbers are marked towards the right from the origin. 
• The negative numbers are marked towards the left from the origin. This is shown in fig.31.3(b)

Fig.31.3

2. Consider another number line. This new number line must be vertical. Let us name it Y'Y. See fig.31.3(b) 
• The positive numbers must be marked upwards from the origin. 
• The negative numbers must be marked downwards from the origin.
3. Now combine the two number lines. 
• The combination should be done in such a way that, the two lines cross each other at their origins.
4. The horizontal line X'X is called the x-axis
• The vertical line Y'Y is called the y-axis 
• The point where X'X and Y'Y cross is called the origin
    ♦ It is denoted by the letter 'O'
5. The positive numbers belonging to the x-axis lies on OX
    ♦ So OX is called the 'positive direction of the x-axis' 
• The negative numbers  belonging to the x-axis lies on OX'
    ♦ So OX' is called the 'negative direction of the x-axis'
• The positive numbers  belonging to the y-axis lies on OY
    ♦ So OY is called the 'positive direction of the y-axis' 
• The negative numbers  belonging to the y-axis lies on OY'
    ♦ So OY' is called the 'negative direction of the y-axis'

6. Consider the fig.31.4 below. Both the x-axis and y-axis are shown. 

Fig.31.4

• The two axes (plural of 'axis' is 'axes') divide the plane of the paper into four parts. 

• These four parts are called quadrants. 

    ♦ In other words, each one of the four parts can be called: A quadrant

(Recall that a polygon with four sides is called a quadrilateral)
7. Each quadrant is given a particular name. The procedure for naming is simple:
• Just number them as I, II, III and IV
• The numbering should be done in the anti clockwise direction starting from OX. So we get the following:
    ♦ Top right: Quadrant I, 
    ♦ Top left: Quadrant II
    ♦ Bottom left: Quadrant III 
    ♦ Bottom right: Quadrant IV
8. Thus the 'plane of the paper' consists of two axes and four quadrants
• This plane is known by three different names. We can use any one of them:
    ♦ The cartesian plane
    ♦ The coordinate plane
    ♦ The xy-plane
9. The axes are called coordinate axes

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Now we will see how the coordinate system can be used to specify the exact location any points on a plane. 

• Consider fig.31.5 below:

Fig.31.5

• Two points P and Q are marked on the cartesian plane. 

• We want to specify the exact locations of P and Q. For that, we use the following steps:

1. Drop the following perpendicular lines:
• perpendicular PM from P on to the x-axis (See fig.31.6 below) 
• perpendicular PN from P on to the y-axis
• perpendicular QR from Q on to the x-axis
• perpendicular QS from Q on to the y-axis

Fig.31.6

2. Dropping the perpendicular lines gives us the following information:
• The perpendicular distance of P from the y-axis is equal to PN. 
• We want the value of this PN. But PN is same as OM (∵ M is the foot of the perpendicular from P)
• We can easily see that OM is 4 units. So PN is 4 units. Thus we get:
• The point P is at a perpendicular distance of 4 units from the  y-axis
    ♦ This 4 units is measured along the positive direction of the x-axis
3. Working in a similar way we get these also:
• The point P is at a perpendicular distance of 3 units from the  x-axis
    ♦ This 3 units is measured along the positive direction of the y-axis
• The point Q is at a perpendicular distance of 6 units from the  y-axis
    ♦ This 6 units is measured along the negative direction of the x-axis
• The point Q is at a perpendicular distance of 2 units from the  x-axis
    ♦ This 2 units is measured along the positive direction of the y-axis
4. Now we have all the required information for specifying the location of the points.
• But there are three rules to be followed. 
• The rules will ensure that the 'final presentation of the results' made by all of us will be in the same form. 
• The 3 rules are:
(i) The x-coordinate of a point, is the perpendicular distance of that point from the y-axis
• This distance should be measured along the x-axis
    ♦ The x-coordinate thus obtained is positive if the measurement is done along OX  
    ♦ The x-coordinate thus obtained is negative if the measurement is done along OX'  
(ii) The y-coordinate of a point, is the perpendicular distance of that point from the x-axis
• This distance should be measured along the y-axis
    ♦ The y-coordinate thus obtained is positive if the measurement is done along OY  
    ♦ The y-coordinate thus obtained is negative if the measurement is done along OY'
(iii) Write the coordinates of a point inside brackets. They must be separated by a comma.
• The x-coordinate is written first and then the y-coordinate
    ♦ Another name for x-coordinate is abscissa
    ♦ Another name for y-coordinate is ordinate
5. Let us apply the rules to point P:
(i) Applying the first rule to find the x-coordinate:
• Perpendicular distance of P from the y-axis (measured along the x-axis) is '4 units'
    ♦ It is measured along OX. So it is positive
    ♦ Thus the x-coordinate of P is '4'
(ii) Applying the second rule to find the y-coordinate:
• Perpendicular distance of P from the x-axis (measured along the y-axis) is '3 units'
    ♦ It is measured along OY. So it is positive
    ♦ Thus the y-coordinate of P is '3'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of P are: (4,3)
6. Let us apply the rules to point Q:
(i) Applying the first rule to find the x-coordinate:
• Perpendicular distance of Q from the y-axis (measured along the x-axis) is '6 units'
    ♦ It is measured along OX'. So it is negative
    ♦ Thus the x-coordinate of P is '-6'
(ii) Applying the second rule to find the y-coordinate:
• Perpendicular distance of Q from the x-axis (measured along the y-axis) is '2 units'
    ♦ It is measured along OY'. So it is negative
    ♦ Thus the y-coordinate of Q is '-2'
(iii) Applying the third rule to write down the coordinates:
• x-coordinate first and then the y-coordinate. Inside brackets and separated by comma. We get:
■ The coordinates of Q are: (-6,-2)
7. In the above example, we drew perpendicular lines PM, PN, QR and QS. If the points P and Q are marked on a graph paper, we will not need to draw those lines. This is shown in the fig.31.7 below:

Fig.31.7

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In the next section we will see some solved examples.

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