In the previous section we saw how to find the coordinates of points in a Cartesian plane. In this section we will see the reverse process. That is., we will be given coordinates of a point. We must locate them on the Cartesian plane. The process of marking points on the Cartesian plane, when their coordinates are given, is called Plotting the points.
Let us see an example:
Given: Coordinates of a point P are (5,3). Plot the point on the Cartesian plane
Solution:
1. The most efficient way for plotting points is to use a graph paper.
2. So, on a graph paper, draw the two coordinate axes: X'X and Y'Y
3. Mark their point of intersection as the origin O
4. Mark the points on the axes choosing a suitable scale. For this problem, '1 cm = 1 unit' will be appropriate. (See details for scale here)
5. Now consider the coordinates. We have '5' as the x-coordinate.
• So the required point will be at a distance of 5 units from the y axis.
6. Draw a line parallel to the y axis. This line should pass through the '5' on the x-axis.
• But on the x axis, there is '5' and '-5'. We must take '5'.
• This is because, 'positive 5' indicates that it is on the positive side of the x-axis. That is., OX
• This line is shown in red colour in fig.31.12 below:
7. It is clear that, the required point lies somewhere on the red line. But where?
• To find that, we consider the y-coordinate. It is '3'
• So the required point is at a distance of 3 units from the x-axis. Also, since it is 'positive 3', the point lies on the positive side of the y axis. That is., OY
8. So draw a line parallel to the x-axis through '3' on OY. This is the green horizontal line in fig.31.12
• It is clear that the required point lies somewhere on the green line. But where?
9. Consider the point of intersection of the red and green lines. That point satisfies both the conditions:
• It is at a distance of 3 5 units from the y-axis
• It is at a distance of 3 units from the x-axis
10. So the point of intersection is our required point P
■ Note that, if we use a graph paper, we will not need to draw the red and green lines because, they will be already present. All we would need to do then, is 'count the number of units' along the axes. This is shown in the fig.31.13 below:
Solved example 31.4
Locate the points (5,0), (0,5), (2,5), (5,2), (-3,5), (-3,-5), (5,-3) and (6,1) in the Cartesian plane
Solution:
• The most efficient way for plotting points is to use a graph paper.
• So, on a graph paper, draw the two coordinate axes: X'X and Y'Y
• Mark their point of intersection as the origin O
• Mark the points on the axes choosing a suitable scale. For this problem, '1 cm = 1 unit' will be appropriate.
A. The given coordinates are (5,0)
1. Draw a vertical line through '5' in OX (OX is used since '5' is positive)
2. Draw a horizontal line through '0'.
• '0' in which one? OX, OX', OY or OY' ?
Ans: There is only one zero. That is at the origin 'O'
• So draw a horizontal through the origin 'O'
• But the horizontal through O is the x-axis
3. So the required lines are:
• The vertical line through '5' in OX
• The x-axis
4. The point of intersection of these two lines will have the coordinates: (5,0)
See fig.31.14 below
B. The given coordinates are (0,5)
1. Draw a vertical line through'0'.
• '0' in which one? OX, OX', OY or OY' ?
Ans: There is only one zero. That is at the origin 'O'
• So draw a vertical line through the origin 'O'
• But the vertical line through O is the y-axis
2. Draw a horizontal line through '5' in OY (OY is used since '5' is positive)
3. So the required lines are:
• The y-axis
• The horizontal line through '5' in OY
4. The point of intersection of these two lines will have the coordinates: (0,5)
See fig.31.14 above
C. The given coordinates are (2,5)
1. Draw a vertical line through '2' in OX (OX is used since '2' is positive)
2. Draw a horizontal line through '5' in OY (OY is used since '5' is positive)
3. The point of intersection of these two lines will have the coordinates: (2,5)
See fig.31.14 above
D. The given coordinates are (5,2)
1. Draw a vertical line through '5' in OX (OX is used since '5' is positive)
2. Draw a horizontal line through '2' in OY (OY is used since '2' is positive)
3. The point of intersection of these two lines will have the coordinates: (5,2)
See fig.31.14 above
E. The given coordinates are (-3,5)
1. Draw a vertical line through '-3' in OX' (OX' is used since '-3' is negative)
2. Draw a horizontal line through '5' in OY (OY is used since '5' is positive)
3. The point of intersection of these two lines will have the coordinates: (-3,5)
See fig.31.14 above
F. The given coordinates are (-3,-5)
1. Draw a vertical line through '-3' in OX' (OX' is used since '-3' is negative)
2. Draw a horizontal line through '-5' in OY' (OY' is used since '-5' is positive)
3. The point of intersection of these two lines will have the coordinates: (-3,-5)
See fig.31.14 above
G. The given coordinates are (5,-3)
1. Draw a vertical line through '5' in OX (OX is used since '5' is positive)
2. Draw a horizontal line through '-3' in OY' (OY' is used since '-3' is positive)
3. The point of intersection of these two lines will have the coordinates: (5,-3)
See fig.31.14 above
H. The given coordinates are (6,1)
1. Draw a vertical line through '6' in OX (OX is used since '6' is positive)
2. Draw a horizontal line through '1' in OY (OY is used since '1' is positive)
3. The point of intersection of these two lines will have the coordinates: (6,1)
• In the above example, we can note the following points:
♦ (5,0) and (0,5) are not at the same position
♦ (-3,5) and (5,-3) are not at the same position
• Several such examples can be shown in a Cartesian plane
■ So in general, we can write:
IF x ≠ y THEN, (x,y) and (y,x) are not at the same position
• That means, we cannot interchange x and y.
• The order of x and y is important in (x,y)
• So (x,y) is called an ordered pair
■ IF x ≠ y, THEN, [ordered pair (x,y)] ≠ [ordered pair (y,x)]
■ IF x = y, THEN, [ordered pair (x,y) = ordered pair (y,x)]
Solved example 31.5
Plot the following ordered pairs (x,y) of numbers as points in the Cartesian plane.
Use scale 1 cm = 1 unit on the axes
Solution:
The points are plotted in the fig.31.15 below:
The reader may make his/her own plot and compare with the above fig.
Solved example 31.6
In which quadrant or on which axis do each of the points (-2,4), (3,-1), (-1,0), (1,2) and (-3,-5) lie? Verify your answers by locating them on the Cartesian plane
Solution:
We can use the following rules:
• Points in the first quadrant will be in the form: (+,+)
• Points in the second quadrant will be in the form: (-,+)
• Points in the third quadrant will be in the form: (-,-)
• Points in the fourth quadrant will be in the form: (+,-)
• Points in the x-axis will be in the form: (±x,0)
• Points in the y-axis will be in the form: (0, ±y)
1. (-2,4) will lie in the second quadrant
2. (3,-1) will lie in the fourth quadrant
3. (-1,0) will lie in the x-axis
4. (1,2) will lie in the first quadrant
5. (-3,-5) will lie in the third quadrant
The plot is shown in the fig.31.16 below:
Solved example 31.7
Plot the following ordered pairs (x,y) of numbers as points in the Cartesian plane.
Use scale 1 cm = 1 unit on the axes
Solution:
The points are plotted in the fig.31.17 below:
The reader may make his/her own plot and compare with the above fig.
Let us see an example:
Given: Coordinates of a point P are (5,3). Plot the point on the Cartesian plane
Solution:
1. The most efficient way for plotting points is to use a graph paper.
2. So, on a graph paper, draw the two coordinate axes: X'X and Y'Y
3. Mark their point of intersection as the origin O
4. Mark the points on the axes choosing a suitable scale. For this problem, '1 cm = 1 unit' will be appropriate. (See details for scale here)
5. Now consider the coordinates. We have '5' as the x-coordinate.
• So the required point will be at a distance of 5 units from the y axis.
6. Draw a line parallel to the y axis. This line should pass through the '5' on the x-axis.
• But on the x axis, there is '5' and '-5'. We must take '5'.
• This is because, 'positive 5' indicates that it is on the positive side of the x-axis. That is., OX
• This line is shown in red colour in fig.31.12 below:
7. It is clear that, the required point lies somewhere on the red line. But where?
• To find that, we consider the y-coordinate. It is '3'
• So the required point is at a distance of 3 units from the x-axis. Also, since it is 'positive 3', the point lies on the positive side of the y axis. That is., OY
8. So draw a line parallel to the x-axis through '3' on OY. This is the green horizontal line in fig.31.12
• It is clear that the required point lies somewhere on the green line. But where?
9. Consider the point of intersection of the red and green lines. That point satisfies both the conditions:
• It is at a distance of 3 5 units from the y-axis
• It is at a distance of 3 units from the x-axis
10. So the point of intersection is our required point P
■ Note that, if we use a graph paper, we will not need to draw the red and green lines because, they will be already present. All we would need to do then, is 'count the number of units' along the axes. This is shown in the fig.31.13 below:
 |
| Fig.31.13 |
Locate the points (5,0), (0,5), (2,5), (5,2), (-3,5), (-3,-5), (5,-3) and (6,1) in the Cartesian plane
Solution:
• The most efficient way for plotting points is to use a graph paper.
• So, on a graph paper, draw the two coordinate axes: X'X and Y'Y
• Mark their point of intersection as the origin O
• Mark the points on the axes choosing a suitable scale. For this problem, '1 cm = 1 unit' will be appropriate.
A. The given coordinates are (5,0)
1. Draw a vertical line through '5' in OX (OX is used since '5' is positive)
2. Draw a horizontal line through '0'.
• '0' in which one? OX, OX', OY or OY' ?
Ans: There is only one zero. That is at the origin 'O'
• So draw a horizontal through the origin 'O'
• But the horizontal through O is the x-axis
3. So the required lines are:
• The vertical line through '5' in OX
• The x-axis
4. The point of intersection of these two lines will have the coordinates: (5,0)
See fig.31.14 below
 |
| Fig.31.14 |
1. Draw a vertical line through'0'.
• '0' in which one? OX, OX', OY or OY' ?
Ans: There is only one zero. That is at the origin 'O'
• So draw a vertical line through the origin 'O'
• But the vertical line through O is the y-axis
2. Draw a horizontal line through '5' in OY (OY is used since '5' is positive)
3. So the required lines are:
• The y-axis
• The horizontal line through '5' in OY
4. The point of intersection of these two lines will have the coordinates: (0,5)
See fig.31.14 above
C. The given coordinates are (2,5)
1. Draw a vertical line through '2' in OX (OX is used since '2' is positive)
2. Draw a horizontal line through '5' in OY (OY is used since '5' is positive)
3. The point of intersection of these two lines will have the coordinates: (2,5)
See fig.31.14 above
D. The given coordinates are (5,2)
1. Draw a vertical line through '5' in OX (OX is used since '5' is positive)
2. Draw a horizontal line through '2' in OY (OY is used since '2' is positive)
3. The point of intersection of these two lines will have the coordinates: (5,2)
See fig.31.14 above
E. The given coordinates are (-3,5)
1. Draw a vertical line through '-3' in OX' (OX' is used since '-3' is negative)
2. Draw a horizontal line through '5' in OY (OY is used since '5' is positive)
3. The point of intersection of these two lines will have the coordinates: (-3,5)
See fig.31.14 above
F. The given coordinates are (-3,-5)
1. Draw a vertical line through '-3' in OX' (OX' is used since '-3' is negative)
2. Draw a horizontal line through '-5' in OY' (OY' is used since '-5' is positive)
3. The point of intersection of these two lines will have the coordinates: (-3,-5)
See fig.31.14 above
G. The given coordinates are (5,-3)
1. Draw a vertical line through '5' in OX (OX is used since '5' is positive)
2. Draw a horizontal line through '-3' in OY' (OY' is used since '-3' is positive)
3. The point of intersection of these two lines will have the coordinates: (5,-3)
See fig.31.14 above
H. The given coordinates are (6,1)
1. Draw a vertical line through '6' in OX (OX is used since '6' is positive)
2. Draw a horizontal line through '1' in OY (OY is used since '1' is positive)
3. The point of intersection of these two lines will have the coordinates: (6,1)
• In the above example, we can note the following points:
♦ (5,0) and (0,5) are not at the same position
♦ (-3,5) and (5,-3) are not at the same position
• Several such examples can be shown in a Cartesian plane
■ So in general, we can write:
IF x ≠ y THEN, (x,y) and (y,x) are not at the same position
• That means, we cannot interchange x and y.
• The order of x and y is important in (x,y)
• So (x,y) is called an ordered pair
■ IF x ≠ y, THEN, [ordered pair (x,y)] ≠ [ordered pair (y,x)]
■ IF x = y, THEN, [ordered pair (x,y) = ordered pair (y,x)]
Solved example 31.5
Plot the following ordered pairs (x,y) of numbers as points in the Cartesian plane.
| x | -3 | 0 | -1 | 4 | 2 |
|---|---|---|---|---|---|
| y | 7 | -3.5 | -3 | 4 | -3 |
Solution:
The points are plotted in the fig.31.15 below:
 |
| Fig.31.15 |
Solved example 31.6
In which quadrant or on which axis do each of the points (-2,4), (3,-1), (-1,0), (1,2) and (-3,-5) lie? Verify your answers by locating them on the Cartesian plane
Solution:
We can use the following rules:
• Points in the first quadrant will be in the form: (+,+)
• Points in the second quadrant will be in the form: (-,+)
• Points in the third quadrant will be in the form: (-,-)
• Points in the fourth quadrant will be in the form: (+,-)
• Points in the x-axis will be in the form: (±x,0)
• Points in the y-axis will be in the form: (0, ±y)
1. (-2,4) will lie in the second quadrant
2. (3,-1) will lie in the fourth quadrant
3. (-1,0) will lie in the x-axis
4. (1,2) will lie in the first quadrant
5. (-3,-5) will lie in the third quadrant
The plot is shown in the fig.31.16 below:
 |
| Fig.31.16 |
Plot the following ordered pairs (x,y) of numbers as points in the Cartesian plane.
| x | -2 | -1 | 0 | 1 | 3 |
|---|---|---|---|---|---|
| y | 8 | 7 | -1.25 | 3 | -1 |
Solution:
The points are plotted in the fig.31.17 below:
 |
| Fig.31.17 |
In the next section we will see some very simple practical applications of the Cartesian plane.
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