TITLE: Lampshade Geometry Problem
QUESTION [45 upvotes]: Today, I encountered a rather interesting problem in a waiting room:
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Notice how the light is being cast on the wall?  There is a curve that defines the boundary between light and shadow.  In my response below, I will prove, algebraically, that this curve is a hyperbola.  I'm also interested in seeing a variety of other solutions, so please feel free to post your own.

REPLY [2 votes]: This is a textbook case of conic sections, I might have even seen a lamp in the actual high school textbook. It's also very obviously a hyperbola because it has two branches. This helps you to see the solution without any computation!
You might argue that having both top and both illuminated parts doesn't prove hyperbola, because the geometry of the light might be different and so it wouldn't mean the parts belong to the same mathematical curve. However, you can always imagine the top lamp to be symmetric in such a way, that the double cone of light really is a double cone (with the same angle and same orientation in both directions). That way even if the physical light is not symmetric, each branch is still a parabola, even if not from the same curve.