Fundamentals of coordinate geometry

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The study of geometry using a coordinate system and the principles of algebra and analysis is called coordinate geometry. Analytic geometry is widely used in physics and engineering and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.

 

The coordinate Plane

In coordinate geometry, points are placed on the "coordinate plane" which is a two-dimensional surface on which points are plotted and located by their x and y coordinates. It has two scales - one running across the plane called the "x-axis" and another at a right angle to it called the y-axis. The point where the axes cross is called the origin and is where both x and y are zero.

 

The coordinate plane

 

X axis

The horizontal scale is called the x-axis. As you go to the right on the scale from zero, the values are positive and get larger. As you go to the left from zero, they get more and more negative.

 

Y axis

The vertical scale is called the y-axis. As you go up from zero the numbers are increasing in a positive direction. As you go down from zero they get more and more negative. A point's location on the plane is given by two numbers,the first tells where it is on the x-axis and the second which tells where it is on the y-axis. Together, they define a single, unique position on the plane. These are the coordinates of the point, sometimes referred to as its "rectangular coordinates". Note that the order is important; the x coordinate is always the first one of the pair.

 

Distance formula

Consider 2 points (x1, y1) and (x2, y2) in the coordinate plane. The distance between them is given by

 

The formula can be understood using Pythagoras theorem.

 

Lines

Straight lines in coordinate geometry are the same idea as in regular geometry, except that they are drawn on a coordinate plane and there are various ways of defining a line and relate it to other figures in the plane.

Lines and their equations in Cartesian plane

 

Definitions

Intercept

The distance from the origin at which the line cuts the coordinate axes is the value of the x and y-intercepts. In the given figure the line 2y= x+1 cuts the y-axis at y=0.5 and x-axis at x= -1. Thus, these two are the intercepts.

 

Slope of a line

 

slope of a line

 

The slope of a line is a number that measures its "steepness", usually denoted by the letter m. It is the change in y for a unit change in x along the line. It is equal to the value of the trigonometric tangent function of the angle made by the line with the x axis. For a line passing through two points (x1, y1) and (x2, y2), the slope is equal to

Two lines which are parallel have the same slope.

Two lines which have the slopes as m1 and m2 are perpendicular is m1 x m2.= -1

The slope of the x axis is 0 and slope of the y-axis is undefined.

 

Equation of a Line

The equation of a line is an equation in which both variables(dependent and independent expressed generally as y and x respectively) have a degree 1 and all points lying on the line satisfy the equation.There are many ways to write the equation of a line.

where, (x1, y1) and (x2, y2) are two points on the line, m is slope and c is the y-intercept.

 

Distance between point and a line

Distance between a point (x1, y1) and a line ax + by + c = 0 can be found as.

The point on this line which is closest to (x1, y1) has coordinates:

       

 

Distance between two parallel lines

The shortest distance between lines can be found by expressing the equations in y=mx+c form, and multiplying the equations by an appropriate factor to make their slopes equal. Let the difference in y-intercept values be d1-d2 and the slope of the lines be m. The distance between the lines is equal to

 

 

 

 


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