Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Calculus or Analysis
    Number Theory
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Easy level:

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 has applications! It 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.

How to Cite this Page:
Su, Francis E., et al. "Reflecting on the Hyperbola." Math Fun Facts. <>.

    Salas and Hille, "Calculus".

Subjects:    geometry, calculus, analysis
Level:    Easy
Fun Fact suggested by:   Michael Moody
Suggestions? Use this form.
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!