This investigation begins with the following problem. Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point. The construction of the circles looks like the following:
Next, we will hide all the lines except for the three circles. Then, we will add the lines through the centers of all the circles. The construction is as follows:
The lines intersect at the center of the constructed triangle. At this point, trace the loci and determine the shape constructed. The trace of the center is the following:
Notice the shape constructed. The center of the constructed circles is connected by segments to the centers of the two given circles, you can tell that the sum of the segments is the same as the sum of the radii of the two given circles. Since the sum is constant, the locus of the center of the tangent circle is an ellipse with foci at the centers of the given circles. Before continuing this investigation, lets first look at the definition of an ellipse. An ellipse is the locus of all points in a plane such that the sum of the distances from two given points in the plane, called foci, is constant. Look at the drawing below:
Points C and D both lie of the ellipse. The line segments containing point C -
foci one and foci one - point D are the same length as foci one - D and D - foci two. If you change the locations of Foci one and two, then the ellipses shape will change. Currently, the major axis goes from left to right, but if you bring the two foci closer together and the distance remains the same, then the major axis will shift to top and bottom.
Lets explore what will change the shape and size of our ellipse we constructed earlier, with the smaller circle remaining inside the larger.
The shape of the ellipse stretches from left to right, which gives it more of an oblong shape. Now, let's make the circle considerably larger, but still remaining inside the larger circle. The smaller circle almost touches the larger circle. How do you think this will change the shape of the ellipse we construct?
The ellipse now stretches top to bottom and appears to look like a circle. Mathematically, what is happening to the loci, when we change the smaller circle? The distance between the foci is not being changed. However, the distance from the foci to points C and D is varying. This is what is changing the shape of the ellipse. Let's test two more possibilities. What if the smaller circle is also tangent to the larger circle and lies partly outside and inside the larger?
Figure 1 shows the case when the smaller circle is tangent. This keeps stretching the graph in the up and down direction. However, when the smaller circle lies partially internal and external with respect to the larger circle, the graph continues to be stretched up/down. An important observation, so far the center of the circle lies internal to the larger circle.
What happens when the smaller of the two given circles (center) is external? We will construct this situation?
Why does this happen? Up to this point the locus of points have constructed a circle. Now, the set of loci constructs a hyperbola!! First, let us revisit the definition of a hyperbola. A hyperbola is the locus of all points in the plane such that the absolute value of the differences of the distance from two given points in the plane, called foci, is constant. That is, F1 and F2 are the foci of a hyperbola and P and Q are any two points on the hyperbola, |PF1-PF2| = |QF1 - QF2|. Points P and Q are loci on the hyperbola.
Since we moved the small circle completely outside the larger circle, this created the situation where the loci of the center of the tangent circle constructs an hyperbola.
During my investigation, I found an interesting case. What if the smaller circle is tangent to the larger circle and lies outside the larger circle? What do you think would happen? After completing the animation, the following curve was formed:
As the constructed circle travels along the larger circle, the locus of points creates a circle equal in size to the larger circle (although it is not the larger now). This is the red circle.
Then, I thought would if the two original circles were the same size and tangent, what would be the locus of points for the constructed circle?
The construction looks like a stretched out ellipse (left and right). An unusual shape.
This investigation has been an unusual one. I would have never thought to utilize two circle centers as foci and explore hyperbolas and ellipses. It is fun to explore new ways to look at mathematics.
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