In the interactive picture below, the radius of the blue circle is controlled by segment AC and the radius of the red circle is controlled by segment CB. So as you drag point C along segment AB, you'll see the two circles grow and shrink in response to the changing lengths of the segments.
Notice that points D and E are the intersection points of the two circles. As you drag point C along segment AB, what curve do these two points seem to trace? To clear the trace, click on the red 'X' in the lower right corner.
You can vary the distance between the circles' centers by dragging either point F1 or F2. You can change the combined length of the circles' radii by dragging point B.
Where are the focal points of your ellipse (assuming that it is one)?
One definition of an ellipse is "the set of points such that the sum of the distances from each point to two fixed points (the foci) is constant." Explain why the construction above satisfies this definition.
The interactive picture below is almost identical to the one above. Only now, point C sits outside segment AB. The blue circle has length AC while the red circle has length BC.
Drag point C back and forth both to the right of AB and to the left of AB. What curve do points D and E seem to trace? You can change the curve by varying the distance between F1 and F2 or dragging point A. When you're done, scroll down.
One definition of a hyperbola is "the set of points such that the difference of the distance from each point to two fixed points (the foci) is constant." Explain why the construction above satisfies this definition.
For a related topic, check out Constant Perimeter Rectangles.