Google Classroom
GeoGebraGeoGebra Classroom

Parabola as Envelope

The parabola is the locus of points in that plane that are equidistant from a line, called directrix and a fixed point, called focus. Another way to view the graph of a parabola is by constructing its envelope, that is the set of all the tangent lines at any point of it. In the following app you can:
  • set the position of the directrix by moving the two diamond points that define it
  • set the position of the focus by dragging point
  • start or pause the animation using the button on the bottom left, or move the green point along the directrix to animate the construction manually
  • delete the trace of the envelope, using the button on top right
The point of intersection of the perpendicular bisector of the segment that joins the moveable point to the focus with the perpendicular line to the directrix is equidistant from the focus and the directrix (why?). When the point moves along the directrix, the perpendicular bisectors define the envelope of the parabola.