Ellipse as Envelope
What happens to a conic if its directrix is not a straight line, but a circle?
The construction method is the same as for the envelope of a parabola, but in this case the directrix of the conic is a circle.
Therefore we need to determine the points that are equidistant from the focus and the circle.
The perpendicular bisector of the segment joining the focus and the point on the circle is the locus of the points equidistant from and . In order to find the points that are also equidistant from the circle, we draw the tangent line to the circle at and the normal line to it.
The intersection of the normal and the perpendicular bisector of segment is a point of the ellipse. When moves along the circle, the perpendicular bisectors of segment will become the envelope of the ellipse.
In the following app you can:
- place point (that is one of the foci of the ellipse) anywhere inside the circle
- start or pause the animation using the button on the bottom left, or move the green point along the circle to animate the construction manually
- delete the trace of the envelope, using the button on top right
The construction is based on point , that we know it's a focus of the ellipse. Where is the other focus?
If you move the focus and make it coincide with the center of the circle, do you still get the envelope of an ellipse? Explain the result and prove it using geometry.