Cartesian System of Rectangular Co-Ordinates, Objectives, Rectangular Co-Ordinate Axes, Cartesian Co-Ordinates of a Point

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  • You must have searched for your seat in a cinema hall, a stadium, or a train. For example, seat means the fourth seat in the row. In other words, and are the coordinates of your seat. Thus, the geometrical concept of location is represented by numbers and alphabets (an algebraic concept).

  • Also a road map gives us the location of various houses (again numbered in a particular sequence), roads and parks in a colony, thus representing algebraic concepts by geometrical figures like straight lines, circles and polygons.

  • The study of that branch of Mathematics which deals with the interrelationship between geometrical and algebraic concepts is called Coordinate Geometry or Cartesian Geometry in honour of the famous French mathematician Rene Descartes.

  • In this lesson we shall study the basics of coordinate geometry and relationship between concept of straight line in geometry and its algebraic representation.

Objectives

After studying this lesson, you will be able to:

  • Define Cartesian System of Coordinates including the origin, coordinate axes, quadrants etc.

  • Derive distance formula and section formula;

  • Derive the formula for area of a triangle with given vertices;

  • Verify the collinearity of three given points;

  • State the meaning of the terms inclination and slope of a line;

  • Find the formula for the slope of a line through two given points;

  • State the condition for parallelism and perpendicularity of lines with given slopes;

  • Find the intercepts made by a line on coordinate axes;

  • Find the angle between two lines when their slopes are given;

  • Find the coordinates of a point when origin is shifted to some other point;

  • Find transformed equation of curve when origin is shifted to another point.

Rectangular Co-Ordinate Axes

  • The fixed point ,where these lines intersect each other is called the origin as shown in Fig.

  • These mutually perpencular lines are called the coordinate axes. The horizontal line is the x-axis or axis of and the vertical line is the y- axis or axis of .

The origin O

The Origin O

The origin O

Cartesian Co-Ordinates of a Point

  • To find the coordinates of a point we proceed as follows. Take and as coordinate axes. Let be any point in this plane. From point draw and .

  • Then the distance measured along x-axis and the distance measured along y-axis determine the position of the point with reference to these axes.

  • The distance measured along the axis of is called the abscissa or x-coordinate and the distance measured along y-axis is called the ordinate or y-coordinate of the point .

  • The abscissa and the ordinate taken together are called the coordinates of the point . Thus, the coordinates of the point are (x and y) which represent the position of the point point in a plane.

  • These two numbers are to form an ordered pair because the order in which we write these numbers is important.

The coordinates of the point P are (x and y)

The Coordinates of the Point P Are (X and Y)

The coordinates of the point P are (x and y)

In Fig. you may note that the position of the ordered pair is different from that of . Thus, we can say that and are two different ordered pairs representing two different points in a plane.

The position of the ordered pair (3,2) and (2,3)

The Position of the Ordered Pair (3,2) and (2,3)

The position of the ordered pair (3,2) and (2,3)

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