# The Normal Form of a Line

If is the length of a perpendicular from origin to the non-vertical line and is the inclination of , then show that the equation of the line is

To prove this equation of a straight is in normal form, let be any point on the straight line . Since the line intersects the coordinate axes at points and , then and become its X-intercept and Y-intercept as shown in the given diagram. Now using the equation of a straight line intercepts form, we have

If is the foot of the perpendicular drawn from origin to the non-vertical straight line, then consider is the right triangle as given in the diagram. So we use the trigonometric ratio as follows:

Since is a right triangle, then

Now putting the values of and in equation (i), we get

This is the equation of a straight line in normal form.