Introduction: Coordinate Geometry

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 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.

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 vertical number line is called Y-axis and the point of intersection of these two axes is known as origin and it is denoted as ‘ O ‘.

Note: 1. Coordinate plane is also known as 2- dimensional plane.

2. X – axis is named as XX’ and Y -axis as YY’

Quadrants: The Coordinate axes XX’ and YY’ divides the cartesian plane into 4 quadrants. In fig. 3 shown below:

- The region XOY is called first quadrant
- The region X’OY is called second quadrant
- The region X’OY’ is called third quadrant
- The region Y’OX is called fourth quadrant

Sign Convention: The ray OX on X-axis is taken as positive, OX’ as negative X-axis, OY on Y-axis as positive and OY’ as negative.

Accordingly, the distance measured along OX will be taken as positive and along OX’ will be negative. Similarly, the distance along OY will be taken as positive and along OY’ will be negative.

I- quadrant (+,+)

II-quadrant (-, +)

III-quadrant(-,-)

IV-quadrant(+,-)

Cartesian Co-ordinates of a point: Consider a point P in a plane.

The length of the line segment \(\overline{OR}\)

Thus, for any given point, the abscissa and ordinate are the distance of a given point from X- axis and Y- axis respectively. The position of point P is given as (x,y).

Take a point on the X-axis, then clearly the distance of the point from X-axis is zero. Thus, the ordinate or Y-coordinate of every point on the X-axis is zero. Hence the coordinate of a point on X-axis is given as (x,0).

Similarly for a point on Y- axis, distance of the point from Y-axis is zero i.e., abscissa is zero. Hence the coordinates of a point on Y-axis is given by(0,y).

Note: 1. If x≠y, then (x,y)≠(y,x) and (x,y) = (y,x), only if x=y .

2. The coordinates of the origin are (0,0).

Application:

- The coordinate plane concept is used in route maps.
- It is also used to locate the position of aircrafts in sky.

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