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Normal form of straight line - CodesJava

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Normal form of straight line


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Coordinate geometry

Straight Lines: Normal form

A normal to a line is considered as a line segment drawn from a point that is perpendicular to the given line.

The normal form of a linear equation of a straight line uses two parameters: p and α to describe the line. Here, p stands for the length of the perpendicular from the origin to the line. And α is the angle between this perpendicular and the x-axis .Let p be the length of the normal drawn from the origin to a line, which subtends an angle α with the +ve direction of the x-axis as follows. Consider the graph below where p is the length of a perpendicular from origin to the non-vertical line l and α is the inclination of p, then show that the equation of the line is

https://www.emathzone.com/wp-content/uploads/2015/01/normal-form-line.png

To prove this equation of a straight line is in normal form, consider P(x,y) be any point on the straight line l. Since the line intersects the coordinate axes at points A and B, then OA and OB become its X-intercept and Y-intercept. Now using the equation of a straight line intercepts form, we have

If C is the foot of the perpendicular drawn from origin O to the non-vertical straight line, then consider OCA is the right triangle as given in the diagram. Use the trigonometric ratios to obtain:

This means that OC =p

Since OCB is a right triangle, then OCOB=cos(90°−α)

This form of the equation is called the normal form.

To convert the general equation of a line into the normal form:

Begin with the general or linear form of the straight line equation:

ax+by+c=0

From the above equations, we obtain:

Cos α = p/m and Sin α = p/n

From Trigonometric identity, we have

Recall the trigonometric identity: Cos2α + Sin2α =1 and the trigonometric ratios:

Image result for triangle sin cos tan

Distance between a point and a line:

Consider the graph below

The distance from point A to a line l is equal to the distance between the given line and the parallel line passing through point A. This means that the length p+d and coordinates of A are sufficient to obtain the line equation:

 

 

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