Jadonna Brewton
Fall 2000
Assignment #7
Problems # 1-8

__Tangent Circles__
Given two circles and a point on one
of the circles, construct a circle that is tangent to the two
circles with one point of tangency being the given point.

**SMALLER CIRCLE INSIDE THE LARGER
CIRCLE**
__Case 1:__
The given point **E** is on the larger circle. The smaller circle
is external to the **tangent
circle **(center at **M**).

__Note:__
The *locus* of the center
**M** of the tangent circle as point **E** travels around
the large circle is an *ellipse* with the centers of the two circles (points **A**
and **C**) as the foci.

**SCRIPT** **GSP SKETCH**

__Case 2:__
The given point **E** is on the larger circle. The smaller circle
is internal to the **tangent
circle** (center at **I**).

__Note:__
The *locus* of the center
**I** of the tangent circle as point **E** travels around
the large circle is an *ellipse* with the centers of the two circles (points **A**
and **C**) as the foci.

**SCRIPT** **GSP SKETCH**

__Case 3:__
The given point **E** is on the smaller circle. The smaller circle
is external to the **tangent
circle **(center at **H**).

__Note:__
The *locus* of the center
**H **of the tangent circle as point **E** travels around
the small circle is an *ellipse* with the centers of the two circles (points **B**
and **D**) as the foci.

**SCRIPT**** GSP SKETCH**

__Case 4:__
The given point **E** is on the smaller circle. The smaller circle
is internal to the **tangent
circle **(center at **H**).

__Note:__
The *locus* of the center
**H** of the tangent circle as point **E** travels around
the small circle is an *ellipse* with the centers of the two circles (points **B**
and **D**) as the foci.

**SCRIPT**** GSP SKETCH**

**INTERSECTING CIRCLES**
__Case 1:__
The given point **E** is on the larger circle. Two tangent
circles are constructed. In the picture below, the small circle
is external to the red
tangent circle (center at **J**) and
internal
to the green tangent circle
(center at **L**).

__Note:__
As point **E** travels around the large circle,

(1) the *locus*
of the center **J** of the red tangent circle is an *ellipse*
with the centers of the two circles (points **A** and **C**)
as the foci.

(2) the *locus*
of the center **L** of the green tangent circle is a *hyperbola *with
the centers of the two circles (points **A** and **C**)
as the foci.

**SCRIPT**** GSP SKETCH**

__Case 2:__
The given point **E** is on the smaller circle. Two tangent
circles are constructed. In the picture below, the large circle
is external to the red
tangent circle (center at **I**) and
internal
to the green tangent circle
(center at **K**).

__Note:__
As point **E** travels around the large circle,

(1) the *locus*
of the center **I** of the red tangent circle is an *ellipse*
with the centers of the two circles (points **A** and **C**)
as the foci.

(2) the *locus*
of the center **K** of the green tangent circle is a *hyperbola *with
the centers of the two circles (points **A** and **C**)
as the foci.

**SCRIPT**** GSP SKETCH**

**DISJOINT CIRCLES**
__Case 1:__
The given point **E** is on the larger circle. Two tangent
circles are constructed. In the picture below, the small circle
is internal to the red
tangent circle (center at **J**) and
external
to the green tangent circle
(center at **L**).

__Note:__
As point **E** travels around the large circle,

(1) the *locus*
of the center **J** of the red tangent circle is a *hyperbola*
with the centers of the two circles (points **A** and **C**)
as the foci.

(2) the *locus*
of the center **L** of the green tangent circle is a *hyperbola *with
the centers of the two circles (points **A** and **C**)
as the foci.

** **
**SCRIPT**** GSP SKETCH**

__Case 2:__
The given point **E** is on the smaller circle. Two tangent
circles are constructed. In the picture below, the large circle
is internal to the red
tangent circle (center at **I**) and
external
to the green tangent circle
(center at **K**).

__Note:__
As point **E** travels around the large circle,

(1) the *locus*
of the center **I** of the red tangent circle is a *hyperbola*
with the centers of the two circles (points **A** and **C**)
as the foci.

(2) the *locus*
of the center **K** of the green tangent circle is a *hyperbola *with
the centers of the two circles (points **A** and **C**)
as the foci.

** **
**SCRIPT**** GSP SKETCH**

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