### Lines (and other items in Analytic Geometry)

Let's review some basic **analytic geometry:** *this is description
of geometric objects by numbers and by algebra.*

The first thing is that we have to pick a *special point*, the
**origin**, from which we'll measure everything else. Then,
implicitly, we need to choose a unit of measure for distances, but
this is indeed usually only *implicit*, so we don't worry about it.

The second step is that **points** are described by *ordered pairs* of numbers: the first of the two numbers tells how far
to the *right* horizontally the point is from the *origin*
(and *negative* means go left instead of right), and the second of
the two numbers tells how far *up* from the origin the point is
(and *negative* means go down instead of up). The first number is
the **horizontal coordinate** and the second is the **vertical
coordinate**. The old-fashioned names *abscissa* and *ordinate*
also are used sometimes.

Often the horizontal coordinate is called the *x-coordinate*, and often the vertical coordinate is called the *y-coordinate*, but the letters $x,y$ can be used for many other
purposes as well, so *don't rely on this labelling!*

The next idea is that *an equation can describe a
curve*. It is important to be a little careful with use of language
here: for example, a correct assertion is

*The set of points $(x,y)$ so that $x^2+y^2=1$ is a
circle*.

It is *not strictly correct* to say that $x^2+y^2=1$ *is* a circle, mostly because *an equation is not a circle*, even
though it may *describe* a circle. And conceivably the $x,y$ might
be being used for something other than horizontal and vertical
coordinates. Still, very often the language is shortened so that the
phrase *‘The set of points $(x,y)$ so that’* is omitted. Just be
careful.

The simplest curves are **lines**. The main things to remember are:

**Slope**of a line is*rise over run*, meaning*vertical change divided by horizontal change*(moving from left to right in the usual coordinate system).- The equation of a line passing through a point $(x_o,y_o)$ and
having slope $m$
can be written (in so-called
**point-slope form**) $$y=m(x-x_o)+y_o\;\;\;\;\;\hbox{or}\;\;\;\;\;y-y_o=m(x-x_o)$$ - The equation of the line passing through two points $(x_1,y_1),
(x_2,y_2)$ can be written (in so-called
**two-point form**) as $$y={y_1-y_2\over x_1-x_2}(x-x_1)+y_1$$ - ...unless $x_1=x_2$, in which case the two points are aligned
vertically, and the line can't be written that way. Instead, the
description
**of a vertical line**through a point with horizontal coordinate $x_1$ is just $$x=x_1$$

Of course, the two-point form can be derived from the point-slope form, since the slope $m$ of a line through two points $(x_1,y_1), (x_2,y_2)$ is that possibly irritating expression which occurs above: $$m={y_1-y_2\over x_1-x_2}$$

And now is maybe a good time to point out that there is
nothing sacred about the horizontal coordinate being called ‘$x$’ and
the vertical coordinate ‘$y$’. Very *often* these *do* happen
to be the names, but it can be otherwise, so just pay attention.

#### Exercises

- Write the equation for the line passing through the two points $(1,2)$ and $(3,8)$.
- Write the equation for the line passing through the two points $(-1,2)$ and $(3,8)$.
- Write the equation for the line passing through the point $(1,2)$ with slope $3$.
- Write the equation for the line passing through the point $(11,-5)$ with slope $-1$.

#### Thread navigation

##### Calculus Refresher

- Previous: Domain of functions
- Next: Elementary limits

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