A whispering gallery is a room designed with an elliptical ceiling such that a person standing at one
focus of the ellipse can whisper and be heard by a person standing at the other focus. A Totoro sitting
at one focus of a whispering gallery, is 2 feet from the nearest wall. A Habanero Ninja is sitting at the
other focus, 8 feet away, whispering to the Totoro. What is the length of this whispering gallery?
How high is its elliptical ceiling at the center? How high is the ceiling right above the Totoro?

Question

A whispering gallery is a room designed with an elliptical ceiling such that a person standing at one 
focus of the ellipse can whisper and be heard by a person standing at the other focus.  A Totoro sitting 
at one focus of a whispering gallery, is 2 feet from the nearest wall.  A Habanero Ninja is sitting at the 
other focus, 8 feet away, whispering to the Totoro.  What is the length of this whispering gallery?  
How high is its elliptical ceiling at the center?  How high is the ceiling right above the Totoro?

kainui · Answer

Try to draw a picture to help yourself locate the foci (plural of focus). Since the problem tells you they can only hear each other if they are standing on a focus and an ellipse only ever has 2 foci, you can locate the distances of foci from each other. A focus is always located nearest to the opposite sides, like in this picture:
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