Distance Equations

Throughout the rest of this exploration, I will refer to the two distance
equations as Distance1 and Distance2, as stated above.

Distance2 = 10 (in green). Notice that the center of the red circle is at (3,4) and the center of the green circle is at (-5,-2), which corresponds to their respective equations.

**B.** Now we will explore the sum **Distance1 + Distance2 = C**,
where **C** is some positive integer. **Figure 2** shows the graph
of **Distance1 + Distance2 = 12** (in blue). By adding the distance equations
together an ellipse is formed. Notice that the foci of the ellipse are at
the centers of the two cirlces.

**Figure 3** shows the graph of **Distance1 + Distance2 = 20** (in
purple). As the value of **C** went from 12 in Figure 2 to 20 in Figure
3, the ellipse grew larger. Also, when **C=20** the ellipse that is formed
crosses through the two points of intersection between the two distance
equations.

**Figure 4 **shows the graph of **Distance1 + Distance2 = 30** (in
olive). Notice that the ellipse is continueing to get larger. This ellipse
is also tangent to the outside of the two circles formed from the two distance
equations.

**Figure 5** is when **Distance1 + Distance2 = 31** (in purple). This
shows that when **C** becomes larger than 30, the ellipses that are formed
completely engulfs the two circles formed in Figure 1.

**Figure 6** shows the graph:

(in purple).

The ellipse has now become very narrow now that **C** is approaching
10. As can be seen in

Figure 7, the closer **C** comes to 10, the more like a straight line
the ellipse looks.

Notice how much this figure needed to be enlarged in order to see that the
ellipse was not a straight line.

Figure 7

**C. **Figures 8 through 13 are an exploration of **Distance1*Distance2
= C**, for some integer **C**.

**Figure 8** is the graph of **Distance1*Distance2 = 24 **(in blue).
For **C** values less than 24, the blue objects become smaller and smaller.

**Figure 9** is when **C = 25**. The two objects are now touching.

**Figure 10 **is when **C = 27. **As the value of **C** is increased,
the two objects become one object and form an ellipse. The larger the value
of **C** gets, the larger the ellipses become.

**Figure 11** is for **C = 100 **(in blue), this is a special case
ellipse because it passes through the two points of intersection of the
two circles from** Figure 1**.

**Figure 12** is for **C = 200**, this forms another ellipse which
is also a special case ellipse because it is tangent to the two circles.

Now as the value of **C** continues to get larger the ellipse becomes
larger and surrounds both circles. This is displayed in **Figure 13**
with **C = 250**.

Distance2 / Distance1 = C

What would the graph look like if we had

**Figure 15** shows the answer to the questioned raised in **Figure
14**. Now the new circle formed is tangent to the larger red circle. Is
this what you had predicted?

Now as **C** increases, what do you expect to happen to the circle?
**Figure 16** shows the graph of **Distance2 / Distance1 = 5**. This
forms a circle that is inside of the red circle. It is not shown here, however,
**Distance1 / Distance2 = 5** will just be the reflection of **Figure
16** through the origin. This will thus form a circle that is inside of
the green circle.

Now when **C** **< 2** , what do you expect to happen to the
circle? As the value of **C** decreases below **2**, the circle that
is formed gets increasing larger than the red or green circle. **Figure
17 **displays the graph **Distance2 / Distance1 = 8/5 or 1.6**. Once
again, **Distance1 / Distance2 = 8/5 or 1.6** will just be the reflection
of **Figure 17** through the origin.

There is one more exploration that is of interest. What happens when
**Distance1 / Distance2 = 1 **or **Distance2 / Distance1 = 1** ? This
is intersting because you get a straight line. As the value of **C**
got closer and closer to **1** the the circles continued to get larger
and larger until at **1** they combine to form a straight line. If **C**
drops below **1**, nothing really new is formed. What happens is the
graphs that we have just dicussed for **Distance2 / Distance1 = C** will
become the ones that we discussed for **Distance1 / Distance2 = C** and
vice-versa. This can be seen with doing some simple math on the equation
**Distance1 / Distance2 = C** or **Distance2 / Distance1 = C**. **Figure
18 **is a graph of the straight line that forms when either **Distance2
/ Distance1 = 1** or **Distance1 / Distance2 = 1**.