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We are familiar with the representation of real numbers on a line, which we call a real line. In this representation we fix a point O (called origin) and represent a real number by a point A on this line such that its distance OA is equal to the value of real number. In the left side of O we represent negative real numbers and in the right side of O we represent positive real numbers. Thus, not only the magnitude of OA but the direction of the line OA is also considered for representation.

Hence OA′ = – AO′

Similarly ordered pairs are represented in a plane. To represent an ordered pair (a, b) we take two reference lines which are mutually perpendicular. The ordered pair (a, b) represents in such a plane, by a point                P(a, b) such that (see figure given below).

OA = a and OB = b.

This system is called Cartesian co-ordinate system. Since elements of an ordered pair are not inter changeable (i.e. (a, b) ≠ (b, a) unless a = b) so they are represented in particular order, the first element ‘a’ is represented on horizontal line called abscissa and the second element ‘b’ on a vertical line called ordinate. Like the real number notation the positive side of the x–axis is the right side of O and positive side of y–axis is upper side of O.

So, the two lines divide the region in 4 parts (See figure). These are called quadrants.

These quadrants are characterized as

I               quadrant                x > 0, y  > 0

II              quadrant                x < 0, y > 0

III             quadrant                x < 0, y < 0

IV             quadrant                x > 0, y < 0

Here the point ‘O’ represents x = 0 and y = 0, hence ordered pair becomes (0,0).

There is a second type of representation called the polar co-ordinate system. In this system a reference is fixed to a line (Called the initial line), and a point called the origin in the system. Any point P is represented by ordered pair (r, θ).

Such that

OP = r; The distance of point from origin and ∟POX  = 0  The angular displacement of line OP from fixed line i.e. the initial line. (in the anticlockwise direction)

Clearly ‘a’ = r cos θ  and ‘b’ = r sin θ (see figure given below)

We can find the distance between two points, then from given three points we should be able to find three sides of a triangle formed by these points. The area of this triangle.

Consider three points P₁, P₂ and P₃ in a plane. Let their co-ordinates be (x₁, y₁), (x₂, y₂) and (x₃, y₃) respectively (see figure)

Area of  ΔP₁P₂P₃ = Area of trapezium AC P₃P₁ – Area of trapezium AB P₂P₁ – Area of trapezium BC P₃P₂

Thus we observe that the area of a triangle is positive when vertices are taken in the anticlockwise direction and negative when the vertices are taken in the clockwise direction.

Important

This form is important. It can be used to find area of a quadrilateral, pentagon, hexagon and polygons.

Þ      If three points P₁, P₂ and P₃ are collinear then the determinant must vanish i.e. the area of triangle formed must be zero.

Several methods have been developed by mathematicians to uniquely locate the position of a point in space. The easiest and most widely used one is the Cartesian Coordinate System, which is based on mutually perpendicular axes. In this chapter you will learn about this systems: locating a point in these systems and finding the equation of line passing through these points.

Note that when we say a line passes through two given points, we are imposing two conditions on this line. The conditions can be imposed in several other manners also. e.g. we may say that a line passes through one point and is perpendicular to another line. These two conditions are sufficient to uniquely know our line. In this chapter you will also learn various manners of imposing conditions and finding the equation of line under those conditions.

Having known a line uniquely in space we shall try to a lot of timings with it e.g. divide it in a given ratio, find distances of other points or lines from it, find the angle, which this line makes with some straight line. We shall also try to see how to find the equation of line in a new coordinate system if the relationship between the new coordinate system with the old is known.

OBJECTIVE

Having known a line uniquely in space we shall try to a lot of timings with it e.g. divide it in a given ratio, find distances of other points or lines from it, find the angle, which this line makes with some straight line. We shall also try to see how to find the equation of line in a new coordinate system if the relationship between the new coordinate system with the old is known. Having known all this we will then try to see how these concepts can be used to solve some problems of conventional Geometry.

PRE-REQUISITE

This chapter requires the following Pre-requisites

Þ       High school knowledge of Geometrical Axioms & Identities.

Þ       Basic knowledge of Lines, Circles, Planes, Triangles etc.

Þ    Elementary algebra