Table of contents  
Introduction to Coordinate Geometry  
Cartesian System  
Coordinates of a Point in Cartesian Plane  
Signs of Coordinates in different Quadrants 
Suppose I put a small dot on a sheet of paper with a pen. Can you locate this dot on the paper if I tell you that the dot is at the lower right corner of the paper?
Now, you are able to see the dot but, can you tell me the exact position of the dot?
You will see that the information given above is not sufficient to fix the position of the dot.
Now, if I tell you that the point is nearly 2 cm away from the bottom line of the paper then this will give some idea but still is not sufficient because this would mean that the point could be anywhere, which is 2 cm away from the bottom line.
Therefore, to fix the position of the dot we have to specify its distance from two fixed lines, the right edge and the bottom line of the paper. Therefore, if I say that the dot is also 1 cm away from the right edge of the paper, then we can easily fix the position of the dot.
We see that position of any object lying in a plane can be represented with the help of two perpendicular lines.
Coordinate geometry is the branch of mathematics where we study the position of an object on a plane with reference to two mutually perpendicular lines in the same plane.
3 on number line is located at a distance of 3 units on the right side of origin 0. Similarly, 3 is located at the same distance from origin but on its left side.
In Cartesian system, two perpendicular lines are used, one of them is horizontal (XX’) and the other is vertical (YY’).
Quadrant
The axes (plural of the word ‘axis’) divide the plane into four parts. These four parts are called the quadrants (1/4), numbered I, II, III and IV anticlockwise from OX.
The plane consists of the axes and the four quadrants. We call the plane, the Cartesian plane, or the coordinate plane, or the xyplane. The axes are called the coordinate axes.
A plane is a flat surface that goes infinitely in both directions.
Example: Point P is on the xaxis and is at a distance of 3 units from the yaxis to its left. Write the coordinates of point P.
Point P is at a distance of 3 units towards left, from yaxis.
Coordinates of point P are (3, 0).
Example: Find distances of points C (3, 2) and D (2, 1) from xaxis and yaxis.
C (3, 2)
Distance from x − axis = 2 units
Distance from y − axis = 3 units
D (2, 1)
Distance from x − axis = 1 units
Distance from yaxis = 2 unit
Example: Locate and write the coordinates of a point:
(a) lying on the xaxis to the left of origin at a distance of 4 units. b) above xaxis lying on the yaxis at a distance of 4 units from the origin.
b) above x axis lying on y axis at a distance of 4 units from origin.
(a) The given point is at a distance of 4 units towards left from the yaxis and at a zero distance from the xaxis. Therefore, the x − coordinate of the point is 4 and the y − coordinate is 0.
Hence, the coordinates of the given point are (4, 0). Coordinates of a point on the xaxis are of the form (x, 0) as every point on the xaxis has zero perpendicular distance from the xaxis.
(b) The given point is at a zero distance from the yaxis at a distance of 4 units from the xaxis. Therefore, the x − coordinate of the point is 0 and the y − coordinate is 4. Hence, the coordinates of the given point are (0, 4). Coordinates of a point on the yaxis are of the form (0, y) as every point on the yaxis has zero perpendicular distance from the yaxis.
Example: Write the quadrant in which each of the following points lie:
(i) (2, 4)
(ii) (1, 4)
(iii) (3, 2)
(i) (2, 4)
Here, x coordinate = 2 and y coordinate = 4
As x coordinate and y coordinate both are negative (x < 0, y < 0) ,the given point lies in III quadrant.
(ii) (1, 4)
Here, x coordinate = 1 and y coordinate = 4
As x coordinate is positive and y coordinate is negative (x > 0, y < 0 ) the given point lies in IV quadrant.
(iii) (3, 2)
Here, x coordinate = 3 and y coordinate = 2
As x coordinate is negative and y coordinate is positive (x < 0, y < 0 ) the given point lies in II quadrant.
Example: If the coordinates of a point M are (2, 9) which can also be expressed as (1 + x, y^{2}) and y > 0, then find in which quadrant do the following points lie: P(y, x), Q(2, x), R(x^{2}, y − 1), S(2x,−3y)
We know,
(2, 9) = (1 + x , y^{2})
∴ 2 = 1 + x ⇒ x = 2 – 1
x = 3
9 = y^{2} ⇒ y = ± 3
Now, it is given that y > 0, so we choose the positive value of y.
So, y = 3
Therefore, x = 3 and y = 3
(i) P (y, x)
P (y, x) = P (3, 3) (∵ y = 3 and x = 3)
As x coordinate is positive and y coordinate is negative (x > 0, y < 0 ) the given point lies in IV quadrant.
(ii) Q (2, x)
Q (2, x) = Q (2, 3) (∵ x = 3)
The x coordinate is positive and y coordinate is negative (x > 0, y < 0 ) so the given point lies in IV quadrant.
(iii) R (x^{2}, y −1)
x^{2 }= (−3)2 = 9; y −1 = 3 – 1 = 2
R (x^{2}, y −1) = (9, 2)
As x coordinate and y coordinate both are positive (x > 0, y > 0) ,the given point lies in I quadrant.
(iv) S (2x, −3y)
2x =2 × (3) = 6; 3y = 3 × 3 = 9
S (2x, −3y)= S (6, 9)
As x coordinate and y coordinate both are negative (x < 0, y < 0),the given point lies in III quadrant.
1. What is the Cartesian system? 
2. How do you find the coordinates of a point in a Cartesian plane? 
3. What are the signs of coordinates in different quadrants? 
4. How do you plot coordinates on a Cartesian plane? 
5. What are some practical applications of coordinate geometry? 
62 videos426 docs102 tests

62 videos426 docs102 tests
