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       #Post#: 104--------------------------------------------------
       REFLECTING ON HYPERBOLA
       By: eba95 Date: July 30, 2010, 6:23 am
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       Reflecting on the Hyperbola
       Most calculus students have learned
       of the "reflecting properties" of the
       parabola and the ellipse. If a "beam
       of light" emanates from the focus of
       a parabola in any direction, and is
       "reflected" from the parabola, it
       subsequently travels in a line parallel
       to the axis of the parabola. For the
       ellipse, a beam emanating from a
       focus is reflected by the curve
       through the other focus.
       Less known is a reflecting property
       of hyperbolae. A beam of light is
       directed at one of the foci (with the
       the curve "between" the source and
       the focus) then it will be reflected by
       the curve through the other focus!
       This property of the hyperbola is
       used in radio direction finding (since
       the difference in signals from two
       towers is constant along
       hyperbolae), and in the construction
       of mirrors inside telescopes (to
       reflect light coming from the
       parabolic mirror to the eyepiece).
       Presentation Suggestions:
       Draw a few pictures to illustrate.
       The Math Behind the Fact:
       If F1 and F2 are the foci of a
       hyperbola, and P is a point on one of
       its branches, elementary geometry
       reveals that the tangent to the curve
       at P bisects the angle F1-P-F2. The
       reflecting property then follows from
       this fact.
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