Coordinate System: Cartesian System: Quadrants of Coordinate System

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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.

French mathematician Rene Descartes put these concepts together 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 below.

Coordinate System

Coordinate System

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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 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 ‘.

Cartesian System

Cartesian System

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Note:

1. The coordinate plane is also known as 2- dimensional plane.

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

Quadrants of Coordinate System

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

Quadrants of Coordinate System

Quadrants of Coordinate System

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1. The region XOY is called first quadrant

2. The region X’OY is called second quadrant

3. The region X’OY’ is called third quadrant

4. 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.

Cartesian Co-ordinates of a point

Cartesian Co-Ordinates of a Point

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The length of the line segment is called the coordinate or abscissa of point P and the length of the line segment OS is called the Y-coordinate or ordinate of point P.

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

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

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

Note:

1. If , then and only if

2. The coordinates of the origin are

Applications

  • The coordinate plane concept is used in route maps.

  • It is also used to locate the position of aircrafts in sky.

Example: There are given some coordinate point. These points are shown on the graph.

Solution:

Points on Graph

Points on Graph

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points on Graph

Points on Graph

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Here point

So, 1 is put on the x – axis and -3 is put on the y – axis as negative.

Point

-2 is put on the negative x – axis and 3 is put on the y – axis.

Point

2 is put on the x – axis and -4 is put on the y – axis as negative.

Point

3 is put on the x -axis and 2 is put on the y – axis.

Point

-1 is put on the x – axis as negative and -3 is put on the y – axis as negative.