# Maths Project on Coordinate Geometry

## Introduction:

Rene Descartes, a French Mathematician introduced "coordinate geometry".It is often referred as "analytical geometry". Descartes initially made two famous discoveries.

They are

1) any ordered pair of numbers can be represented by a point in a plane.

2) any curve in the two-dimensional space can be identified by an equation in x and y.

Thecoordinate plane forms the basis of coordinate geometry. The co-ordinate plane is a two-dimensional plane in terms of two axes: x andy. The x-axis indicates the horizontal axis while the y-axis indicates the vertical axis of the plane. The x-axis and y-axis are perpendicular to each other and meet at a point called "origin" .In the coordinate plane, points are indicated by ordered pairs in which first coordinate is x- coordinate also called "abscissa" and second coordinate is y-coordinate also called "ordinate".

## Concepts of coordinate geometry:

Slope:

On the coordinate plane, the slant of a line is called the slope. Slope is the ratio of the change in the y-value over the change in the x-value. Given any two points on a line, you can calculate the slope ofthe line by using this formula:

slope =

For ex:

Given two points, P = (0, –1) and Q = (4,1), on the line we can calculate the slope of the line.

slope

•    Slope of x-axis is 0 and slope of any line parallel to x-axis is zero .
•    Slope of y-axis is not defined and slope of any line parallel to y-axis is also undefined.

General Form of Line Equations:

•   Slope-Intercept Form:

In coordinate geometry, the equation of a line can be written in the form, y = mx + b, where m is the slope and b is the y-intercept.

y = mx+b

Ex: For the equation, slope is 1/2 and y-intercept is -1.

• Point Slope Form:

The equation of the line passing through a given point (x1,y1) and having slope 'm' is:

y - y1 = m (x - x1).

Slope of Parallel Lines:

Two lines are said to be parallel if they have same slope.

Example: The line is parallel to the line because slopes (1/2) are same.

Slope of Perpendicular Lines:

Two lines are said to be perpendicular if the product of their slopes is -1.

Example: The line is perpendicular to the line y = –2x – 1because the product of the two slopes is .

## Mid-point formula and distance formula of coordinate geometry:

Mid-Point Formula:

Mid point that is halfway between two given points, get the average ofthe x-values and the average of the y-values.

The midpoint between the two points (x1,y1) and (x2,y2) is

Ex: The midpoint of the points A(1,4) and B(5,6) is
.

Distance Formula:

The distance between the two points (x1,y1) and (x2,y2) is

Ex: Distance between A(1,1) and B(3,4) is

2 = 22 + 32
2 = 13

= .