The Meaning of Locus

Date: 03/04/2003 at 20:18:01
From: Marina
Subject: Locus

What is the locus of points equidistant from two parallel lines 8 
meters apart?

I have trouble understanding the meaning of locus.

Date: 03/04/2003 at 23:14:54
From: Doctor Peterson
Subject: Re: Locus

Hi, Marina.

The idea of locus is very simple, but a little subtle. You can almost 
leave the word out of many problems; this one asks you simply to 
describe all points that are equidistant from the two lines. The 
point of the word "locus" is merely that you are to think of all the 
points that fit that description as a single entity, which might be a 
curve or line or a set of discrete points.

So picture it one point at a time. What sort of point is the same 
distance from both lines? Imagine a point somewhere; how do you 
measure the distance from each line? You draw a perpendicular from 
the point to each line, and their lengths are the two distances. Now, 
what does it mean if they are the same? You will find that the points 
you are looking for are exactly between the two lines.

Once you get that idea, think about how you can describe all such 
points. Imagine that you were forced somehow to walk only in places 
that are the same distance from the two lines, perhaps because there 
is a rope from you to each of the lines and some mechanism keeps them 
the same length. Where will the grass be trampled down? That is the 
locus.

If you need more help, please write back and tell me what you found.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/