The Locus of tangent Circles
If we were to trace the locus of two tangent circles, as presented, we obtain an ellipse.
we want to observe what happens when the circles become secant.
Though the circles have relocated, we notice that the centers, which are the foci, of the circles remain inside the locus. To investigate further, what happens when the circle B is just tangent to circle A?
As you can see, the circle of the locus merges with the circle of A so that it is still tangent to B. Also, as the circles A and B draw further away from each other, the ellipse stretches. An ellipse is made of two axes, a major axis and a minor axis. The major axis is constructed as the line segment between the foci as the circles are moving from each other.
However, when the circles are totally separated from each other, we have a total different figure by the locus.
Suddenly, the locus is no longer an ellipse. We now have a hyperbola, and the hyperbola continues to infinity as the distance of the foci approaches infinity.