Thursday, July 18, 2013

Another Interesting Curve - The Ellipse

One of the topics that is often studied in pre-calculus is conic sections.  It is great to be able to give students examples of things that are actually shaped like one of the conic sections.

Today I want to focus on the ellipse.

The definition of an ellipse is "a locus of points such that the sum of the distances from two other points (the foci) is constant."

As you can see from the picture above, you can have students draw an ellipse by tacking two down two ends of a piece of string.  Then, students should put a pencil in the string as shown.  Carefully they move the pencil around the piece of string and they will trace out an ellipse.  That makes sense because as we see in the definition, the length of the string will remain constant.

But, the interesting part...where can we see ellipses in our lives?  

1)  Here is a picture of a park that is in the shape of an ellipse.  This park is in Washington DC and sometimes called President's Park South.  If you look carefully, you can see the White House at the top of the picture.  The National Christmas Tree is in this park.  This tree is lit every year by the President.

2)  We can also see ellipses in the orbits of the planets around the sun.  Although people once thought that planets orbited the sun in a circular path, we now know that the path is an ellipse.  Kepler's First Law says that planets orbit the sun in the shape of an ellipse with the sun as one of the foci (a focus is one of the fixed points that we mentioned in the definition above.)

3) The ellipse has a reflective property...If a tangent line is drawn at point P, the angles formed by that line and the lines PF' and PF would be equal. So, a sound wave that starts at one of the foci is reflected back to the other foci...this is called a "whispering gallery."  If two people stand at the foci, they can whisper back and forth to each other.

We went on a summer out to the Museum of Science and Industry in Chicago and visited their Whispering Gallery.

Have a great day!

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