Explorations With Graphs and Equations

# By Donna Greenwood

Consider two points (3, 4) and (-5, -2).  For any point (x, y) we can write the distance as

Distance 1 =

Distance 2 =

Explore graphs with these two distance equations.  For example,

1. Consider when each set is set to a non-zero constant.   Circles are graphed.

In this drawing, the distances are set to several different constants as follows:

 Color Distance Blue 3 Aqua 4 Red 5 Magenta 10

It looks like the circles are tangent when the distance from each point is 5.  Is that true?  Looking at the distance between the two points, , equals 10, so the red circles are indeed tangent.  Since the distance between the points is 10, it makes sense that circles graphed with a distance smaller than 5 do not intersect, while those with a distance greater than 5 do.

1. Consider the sum C =

For different values of C.

 Color Value of C Magenta 10 Blue 15 Green 20 Aqua 25 Red 30

As the constant distance C gets larger, the graph appears to approach a circle.

Next, consider product as a constant distance:

 Color Value of C Magenta 10 Green 20 Aqua 25 Red 30 Blue 50

## Here is the graph when the difference of the distances is held constant:

Finally, here is the graph of the division of the distances: