**Assignment #7**

**Investigating
a Tangent Circle**

**by**

**Kimberly
Burrell**

**In this exploration, we begin by
constructing a circle tangent to two given circles with one point
of tangency being designated.**

**The following illustration shows the
two given circles in the blue and the tangent circle in red. In
this case, one of the given circles is completely inside the
other.**

**Now, lets look at the tangent circle (red)
when the given circles intersect.**

**In this illustration, one observes that the
designated point has moved from the previous example. The tangent
circle moves as this point moves.**

**Another case occurs when the two given
circles are disjoint.**

**In this interesting case, we see that one
the given circles is now in the interior of the tangent circle.**

**At this time in our exploration, we may
wonder that the locus (purple) of the center of the tangent
circle looks like. In the first case, with one of the given
circles completely inside the other.**

**One can observe that this locus is also
an ellipse and that the centers of the given circles are the foci
of the ellipse.**

**Now we wonder what the locus is when the
circles are intersecting as in our second case.**

**One can quickly observe that the locus is
also an ellipse and the centers of two given circles are the foci
of the ellipse.**

**Finally, we will explore the situation shown
in our third case with the two given circles being disjoint. ****Click here**** to see
the exploration.**

**In this instance, we see that the locus of
the center of the tangent circle is a hyperbola with the centers
of the two given circles acting as the foci.**

**To explore the pedalogical
value of this assignment, ****click here****.**