One of the things I find most interesting about conic sections is their reflective properties. Here are those properties, as I understand them (for the sake of a common description, I'll suppose a "beam" is bouncing off each conic section):

1) Any beam parallel to the axis of a parabola will be reflected to its focus. Or, any beam leaving the focus of a parabola will be reflected to a path that is parallel to the axis.

2) Any beam that passes through or leaves one focus of an ellipse will be reflected to pass through the other focus.

3) Any beam directed toward one focus of a hyperbola will be reflected toward the other focus.

These are a nice talking point when teaching students about conic sections, since they provide some interesting reasons for locating the foci or finding equations for sections with specific foci or focal distances. Flashlights and satellite dishes are familiar objects for kids, and it's always nice to have an answer to the question "Who uses this stuff?"

One of these days I'd like to team up with the engineering teacher at my school and make a model that demonstrates these properties. I imagine setting up pieces of conics with shared foci, as sketched out below. Our technical education department has a computerized lathe, so I think we could get the appropriate grooves cut into a piece of particle board and then put a thin strip of reflective material in each one. There would be a track for a laser pointer that runs perpendicular to the axis of the parabola, so the beam could move back and forth and always be parallel to the axis. If it were set up correctly, the beam would always hit the last focus (the second focus of the hyperbola) even as the laser pointer is moved back and forth. I'm not sure how tight the precision would have to be, but I guess that last focus could just be as big as necessary.

Maybe students wouldn't get as excited about this as I would, but I think it would be a pretty cool demonstration for class. Here's my attempt at creating the same in GeoGebra, but I'm not sure how to plot only part of a conic section yet.

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