What happens to a conic if its [i]directrix [/i]is not a straight line, but a circle?[br][br]The construction method is the same as for the [url=https://www.geogebra.org/m/h4mpdre8]envelope of a parabola[/url], but in this case the directrix of the conic is a circle. [br]Therefore we need to determine the points that are equidistant from the focus [math]F[/math] and the circle.[br]The perpendicular bisector of the segment joining the focus [math]F[/math] and the point [math]A[/math] on the circle is the locus of the points equidistant from [math]F[/math] and [math]A[/math]. In order to find the points that are also equidistant from the circle, we draw the tangent line to the circle at [math]A[/math] and the normal line to it. [br]The intersection of the normal and the perpendicular bisector of segment [math]AF[/math] is a point of the ellipse. When [math]A[/math] moves along the circle, the perpendicular bisectors of segment [math]AF[/math] will become the envelope of the ellipse.[br][br]In the following app you can:[br][list][*]place point [math]F[/math] (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[/*][/list]