Department of Mathematics Education

J. Wilson, EMAT 6680

 

Matthew Tanner

EMAT 6680

Write-up #7

July 22, 2004

 

 

 

Tangent Circles

 

Associated with any two circles on a plane, there is a family of tangent circles each member of which is uniquely determined by a tangent point on either of the circles.

 

Figure 1

When the interior of either circle lies within that of other, there is a circle for every tangent point and the locus of the centers of the tangent circles forms a continuous closed curve.

 

Figure 2

When the interiors of each circle are distinct, the curve is discontinuous the discontinuities corresponding to tangent lines for which no tangent circle is defined. You can think of them as circles with infinite diameters.

 

 

The graph of the locust of the centers of tangent circles are consistent with the graph of conic sections in the former case that of an ellipse and in the later, a hyperbola.

 

Note There are two cases of tangent circles. For the purpose of illustration and labeling one in which, were one principle circle interior to the other, the tangent circle would be exterior to the interior principle circle the second case in which the tangent circle encompasses the interior principle circle. The general form of this proof is valid for both cases.

 

Figure 3

Recall that an ellipse is the locus of points for which the sum of the length of adjunct line segments emanating from two foci is a fixed constant.

 

Figure 4

To show that at the locus of the centers of the tangent circles, in the former case, is and ellipse, observe that Segment 1 is the sum of the radii of Circle 1 and the Tangent Circle. Segment 2 is the length of the radius of Circle 2 minus the radius of the Tangent Circle.

 

So

 

Segment 1 + Segment 2 =

 

Radius of Circle 1 + Radius of Tangent Circle

+

Radius of Circle 2 - Radius of Tangent Circle

 

=

 

Radius of Circle 1 + Radius of Circle 2.


Since the length of the radii is fixed, we have shown that the curve of the locus of the centers of the tangent circles is an ellipse.

 

Figure 5

To show that the curve in the latter case is a hyperbola, recall that a hyperbola is the locus of all points for which the difference of the length of adjunct line segments emanating from two foci is a fixed constant.

Figure 6

Observe that Segment 1 is again the sum of the radii of Circle 1 and the Tangent Circle. The diameter of Segment 2, in this case however, is the radius of the Tangent Circle minus the radius of Circle 2.

 

So in this case

 

Segment 1 + Segment 2 =

 

Radius of Circle 1 + Radius of Tangent Circle

+

Radius of Tangent Circle -Radius of Circle 2

 

=

 

Radius of Circle 1 - Radius of Circle 2.


Since the length of the radii is fixed, we have shown that the curve of the locus of the centers of the tangent circles in the latter case is a hyperbola.