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Normalised lines - Solution to the problems

 

A look at the normalised form.  We have:

ax+by+c=0

Rewrite it in a form where y is a function of x:  subtract ax+c from both sides, then divide by b.

y = -(a/b)x -(c/b)

This is back to the form y=ax+b where a, the coefficient of x, was the slope.  So we have that the slope is

-a/b

And in that same form b was the height where the line cut the vertical or y axis, so we have that

-c/b

is the height where the line cuts the vertical axis.

Where does it cut the horizontal axis?  That is a point where y=0, since all points on the horizontal axis have y=0.  If we fill that into ax+by+c=0 we get

ax+c=0

from which we get

x = -c/a

Of course, there are problems if either a or b is zero.  If b is zero then the slope makes no sense because we can't divide by zero, but we do know that in that case the line is vertical anyway.

If a is zero the line is horizontal and so it makes no sense to talk about where it cuts the horizontal axis since the line and the axis are parallel.

Beauty

The nice thing about the normalised form is that it looks very symmetric: neither x nor y have the "best role".  Better even, there are no exceptions, all possible lines can be written in that form.

Terminology

The letters x and y are the coordinates of a point, they are called the variables.

The letters a, b and c are numbers.  They are the coefficients of the formula.  They are also the numbers that determine where the line goes, and for that reason they are called the parameters.

It is best before going on to read the page about the difference between formulae, equations and identities.

Then come back here and go on.


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next planned revision: 2009-01