**Pedagogical
Value to the concept of**

**Tangent
Circles**

**The current high school mathematics
curriculum calls for students to study and comprehend the
underlying principles of conic sections including ellipses and
hyperbolas. As we have seen, the locus of the center of the
tangent circle to two given circles is either an ellipse or a
hyperbola, depending on the placement of the given circles.**

**The
Ellipse**

**To begin to teach
students about ellipses, we start off with a discussion of the
following picture, in which F1 and F2 are the foci of the ellipse
and P is any point on the ellipse.**

**P*F1 + P*F2 = k where k
is some constant.**

**It is also important for students to
have a deeper understanding of how to generate the above
illustration and why the equation is true. The construction of a
circle tangent to two given circles clearly demonstrates both of
these concepts.**

**Here one can see that F1 is the center
of one of the given circles and F2 is the center of the other
given circle. P is again any point on the ellipse, and is also
the center of the tangent circle.**

**The
Hyperbola**

**To begin to teach students about
hyperbolas, we start off with a discussion of the following
picture in which F1 and F2 are the foci of the hyperbola and P is
any point on the hyperbola.**

**P*F1 - P*F2 = k where k is some
constant. ****Click here**** ****to
see the picture.**

**Here one can observe that F1 is the
center of one of the given circles and F2 is the center of the
other given circle. P is again any point on the hyperbola, and is
also the center of the tangent circle.**