Notes of Coordinate Geometry

Cartesian System

Cartesian plane & Coordinate Axes

Cartesian Plane: A cartesian plane is defined by two perpendicular number lines, A horizontal line(x−axis)  and a vertical line (y−axis).

These lines are called coordinate axes. (Cartesian plane is named after French mathematician Rene Descartes, who formalized its use in mathematics) The Cartesian plane extends infinitely in all directions.

Origin: The coordinate axes intersect each other at right angles, The point of intersection of these two axes is called Origin.

Two perpendicular number lines intersecting at point zero are called coordinate axes. The horizontal number line is the x-axis (denoted by X’OX) and the vertical one is the y-axis (denoted by Y’OY). The point of intersection of x-axis and y-axis is called origin and denoted by ‘O’. 

Cartesian plane is a plane obtained by putting the coordinate axes perpendicular to each other in the plane. It is also called coordinate plane or xy plane. 

The x-coordinate of a point is its perpendicular distance from y-axis. 

The y-coordinate of a point is its perpendicular distance from x-axis. 

The point where the x axis and the y axis intersect is represented by coordinate points (0, 0) and is called the origin

Quadrants

The cartesian plane is divided into four equal parts, called quadrants. These are named in order as I,II,III and IV starting with the upper right and going around in anticlockwise direction. 

 The abscissa of a point is the x-coordinate of the point. The ordinate of a point is the y-coordinate of the point. 

If the abscissa of a point is x and the ordinate of the point is y, then (x, y) are called the coordinates of the point. 

Quadrants

The axes divide the Cartesian plane into four parts called the quadrants (one fourth part), numbered I, II, III and IV anticlockwise from OX

Sign of coordinates depicts the quadrant in which it lies. 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. 

Quadrants2

The coordinates of a point on the x-axis are of the form (x, 0) and that of the point on y-axis are (0, y). 

To plot a point P (3, 4) in the Cartesian plane, start from origin and count 3 units on the positive x axis then move 4 units towards positive y axis. The point at which we will arrive will be the point P (3, 4). 

Quadrants points

Quadrants

Points in different Quadrants.

Signs of coordinates of points in different quadrants:

  1. Quadrant: ‘+’ x – coordinate and ‘+’ y – coordinate. E.g. (2,3)
  2. Quadrant: ‘-’ x – coordinate and ‘+’ y – coordinate. E.g. (-1,4)
  3. Quadrant: ‘-’ x – coordinate and ‘-’ y – coordinate. E.g. (-3,-5)
  4. Quadrant: ‘+’ x – coordinate and ‘-’ y – coordinate. E.g. (6,-1)

Plotting on a Graph

Representation of a point on Cartesian plane

Using the co-ordinate axes, we can describe any point in the plane using an ordered pair of numbers. A point A is represented by an ordered pair (x,y) where, x is the abscissa and y is the ordinate of the point.

Position of points

Position of a point in a plane

Plotting a point

The location of a point in the plane is given by its coordinates,the first number x gives the point’s horizontal position and the second number y gives its vertical position.

For example, Point (3,2) is 3 units away from positive y-axis and 2 units away from positive x-axis.

Therefore, point (3,2) can be plotted as shown below. Similarly, (-2,3), (-1,-2) and (2,-3) are plotted.

Plotting

Plotting a point in the plane