**Explorations With Graphs and Equations**

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,

- 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.

- 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 |

** **

** **

** **

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