Cartesian Plane

Cartesian Plane

Introduction     
     Mathematics in ancient days was divided into two branches ‘Algebra’ and ‘Geometry’. Algebraic equations were not used in geometry and geometrical figures were not used in algebra. But these two branches were put together by the French mathematician Rene Descartes for the first time. He introduced the concept of the Cartesian plane or coordinate system to explain geometry and algebra together
     The number line is a straight line where the integers are placed at equal distances. All positive numbers are placed on the right-hand side of zero and all negative numbers are placed on the left-hand side of zero as shown in fig. 1.

       When two number lines are placed mutually perpendicular to each other it forms coordinate axes.

Cartesian System
A Cartesian coordinate system or Coordinate system is used to locate the position of any point and that point can be plotted as an ordered pair (x, y) known as Coordinates. The horizontal number line is called X-axis and the vertical number line is called Y-axis and the point of intersection of these two axes is known as the origin and it is denoted as ‘O’.

Note:
1. The coordinate plane is also known as the 2- dimensional plane.
2. X-axis is named as XX’ and Y-axis as YY’.

1.     The origin: The point of intersection of the X-axis and the Y-axis is called the origin.
2.     The coordinate of the origin are (0, 0).
Proof : Let the coordinate of the origin be (x, y)
We know
x = The perpendicular distance of the origin from the Y-axis.
   = 0
y  = The perpendicular distance of the origin from the X-axis 
    = 0
Hence, the coordinates of the origin are (0, 0).
3. The coordinates of the points on the X -axis and the Y-axis.
    The Y-coordinate of the every point on the X-axis is zero since, the perpendicular distance of the point on X-axis from X-axis is zero.
    Similarly, the x-coordinate of the every point on the Y-axis is zero, since the perpendicular distance of the point on Y-axis from Y-axis is zero.

Cartesian Plane

Cartesian Plane

Quadrants : The X-axis and the Y-axis divides the plane in four independent regions called as 'Quadrants'.

4.    The coordinates of various points in the quadrants.


NOTE: (i) \( \left( {x,y} \right) \in Q_1 \Leftrightarrow x > 0,y > 0 \)
    (ii) \( \left( {x,y} \right) \in Q_2 \Leftrightarrow x < 0,y > 0 \)
    (iii) \( \left( {x,y} \right) \in Q_3 \Leftrightarrow x < 0,y > 0 \)
    (iv) \( \left( {x,y} \right) \in Q_4 \Leftrightarrow x > 0,y < 0 \)
    (v) \( \left( {x,y} \right) \in positive\text{ X - axis} \Leftrightarrow y = 0 \) and \( x \in R^ + \)
    (vi) \( \left( {x,y} \right) \in negative\text{ X - axis} \Leftrightarrow y = \text{0 and x} \in R^ - \)
      (vii) \( \left( {x,y} \right) \in \text{X - axis} \Leftrightarrow y = \text{0 and x} \in R \) 
    (viii)\( \left( {x,y} \right) \in positive\text{ Y - axis} \Leftrightarrow x = \text{0 and y} \in R^ + \) 
    (xi) \( \left( {x,y} \right) \in positive\text{ Y - axis} \Leftrightarrow x = \text{0 and y} \in R^ - \)
    (x) \( \left( {x,y} \right) \in \text{ Y - axis} \Leftrightarrow x = \text{0 and y} \in R \) 
    (xi) \( \left( {x,y} \right) \in The\text{ }origin \Leftrightarrow x = \text{0 and y = 0} \)
5.  A point which belongs to either of the coordinate axes does not belongs to any of the quadrants and viceversa.
6.  The origin does not belongs to any of  the quadrants.
7.   The origin belongs to both the coordinate axes i.e., the X-axis and the Y-axis.
8.   The distance between the two points (x, 0) & (x₂, 0) is |x₁ - x₂| or |x₂ - x₁|
9.   The distance between the two points (0, y₁) & (0, y₂) is |y₁ - y₂| or |y₂ - y₁|
10.  The distance between the two points (x₁, y) &  (x₂, y) is |x₁ - x₂| or |x₂-x₁|
11.  The distance between the two points (x, y₁) & (x, y₂) is |y₁-y₂| or |y₂-y₁|
12.  The mid point of the line joining A(x₁, 0) & B(x₂, 0) is \( \left( {\frac{{x_1 + x_2 }} {2},0} \right) \) 
13.  The mid point of the line segment joining A(0, y₁) & B(0, y₂) is \( \left( {0,\frac{{y_1 + y_2 }} {2}} \right) \)