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 (x_{1},y_{1}) and having slope 'm' is:
y - y_{1} = m (x - x_{1}).
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 (x_{1},y_{1}) and (x_{2},y_{2}) is
Ex: The midpoint of the points A(1,4) and B(5,6) is
.
Distance Formula:
The distance between the two points (x_{1},y_{1}) and (x_{2},y_{2}) is
Ex: Distance between A(1,1) and B(3,4) is
^{2} = 2^{2} + 3^{2}
^{2} = 13
= .