In this paper, we consider a locus of points in the plane determined
by some relation to two fixed points. The word *locus*(plural
*loci*) is Latin and translates into English as place. So
if we say we are looking for the locus of points in a plane that
are 5 units from the origin, we are just saying we want to find
all the places that are 5 units away from the origin and describe
what they look like as a group. We say *the locus of points
in a plane 5 units from the origin is a circle with radius 5 centered
at the origin. *The phrase *in a plane *is important,
since without that restriction, (i.e. allowing points in space)
we would be describing a sphere with radius
5 centered at the origin.

For example, consider points A(-3 , 0) and B(3 , 0) on a Cartesian coordinate system. What is the locus of points in the plane equidistant from A and B? We picture points A and B and try to imagine placing points on the page that are the same distance from point A as they are from point B.

The result is a collection of points on a line that is the perpendicular bisector of segment AB. In our specific case, the locus is the y-axis.

What is the locus of points in a plane where the

Given fixed points (a,b) and (c,d) we seek all points (x,y) such that

where k is some chosen constant.

For (a,b) = (-3,0) and (c,d) = (3,0) we obtain the graph shown
below. Click on the graph to open Graphing Calculator and explore
for various values of k, or click **here**
to see a Quicktime movie as k varies from 6 to 20 in steps of
0.1.

So the locus of points in a plane chosen so that the sum of the distances to two given points is constant is an ellipse, as we suspected since this is the locus definition of ellipse. The given points are the foci of the ellipse.

What if we ask for the locus of points in the plane such that the

We started with the locus definition of a hyperbola, and as you can see, the locus of points appears to be a hyperbola and the two given points are the foci.

So far, our discussions have not really deviated from the typical
high school curriculum. But what if we ask for the locus of points
in a plane such that the ** product** of the distances
to two fixed points is a constant? Simply changing the addition
or subtraction to a multiplication in one of the above investigations
gives an immediate visual response to our question. As before
click anywhere on the graph to open Graphing Calculator, or click

This investigation is easily within the grasp of a high school student, who should be asked to find the value of k for which the graph degenerates into two pieces and to explore algebraically why it happens.

It is interesting to think of the graph for each value of k
as a slice of a three-dimensional figure. Click **here**
to see a Quicktime movie or **here**
to open Graphing Calculator and explore this 3-D graph.

Suggested topics for Web research are **lemniscates** and
**ovals of Cassini.**

Of course, any of these figures may be rotated by selecting the fixed points not on the axes - another topic that often gets little attention in high school mathematics, but is easily accomplished with graphing software.

An obvious question for any of the above investigations is this. Why limit the discussion to two points in a plane? Why not three? One of these is presented here. What is the locus of points in a plane such that the product of the distances to

When I started this web page, I considered including the question
*What is the locus of points in a plane such that the ratio
of the distances to two fixed points is a constant?*, but I
didn't think it was very interesting since it was obvious the
locus was a circle. Then in the December, 2001 issue of

Click **here** to see my proof.
To see this locus definition of circle implemented in GSP, click
**here**.

Email to dhembree@coe.uga.edu

The title of this write-up "Stuck in loci again"
is a play on a lyric from a Credence Clearwater Revival song called
*Lodi.*