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.

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SCRIPT                                                 GSP SKETCH

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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.

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SCRIPT                                             GSP SKETCH
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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.

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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.

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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.

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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.

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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.

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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.

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SCRIPT                                      GSP SKETCH```

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