Doug Ensley's Envelope Applet

In the window below, you can explore envelopes of families of lines.  The interaction is a Flash applet by Doug Ensley, created as part of the Flash and Math Project.   Each family of lines is defined by an equation of the form

x/a(p) + y/b(p) = 1

where a(p) and b(p) are functions of the parameter p.  For any value of p, the equation produces a straight line whose x and y intercepts are
a(p) and b(p), respectively.  Varying p produces a collection of lines -- the family of lines for the given equation.  In the applet you can specify a(p) and b(p) and then generate a sample of lines from the family.  You can also display the envelope for the family of lines.   Can you find the equation of the envelope?  The applet permits you to specify a solution, either implicitly or as a parameterized curve, and will add your solution to the graph.

To begin immediately, scroll down to the applet window.  Or if you prefer, view some more detailed instructions first.  You may also wish to read more about envelopes and methods for finding their equations.