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Definition of an Ellipse

Date: 4 Jan 1995 11:00:53 -0500
From: Denise Lu
Subject: (none)

To Whom it May Concern:

I have a question concerning the concept of an ellipse. It is said that 
the equation for an ellipse is Pf + Pr= 2a where P is a point on the 
ellipse and f and r are the points of the foci. How do we know that this 
is true, that is that Pf + Pr = 2a? How did we come up with the constant 
of 2a?  
                                                  Thank you,
                                                  Denise Lu 

Date: 4 Jan 1995 17:26:02 -0500
From: Dr. Ken
Subject: Re: your mail

Hello there!

As a matter of fact, this equation is the _definition_ of an ellipse
(although different people will give you different definitions, this is
certainly a standard one).  Let's see what it means.

You know that a circle is all the points in a plane which are a constant
distance from a fixed point.  We call this distance the radius of the
circle.  Well, an ellipse is kind of a more general form of the circle.  
We define an ellipse as the set of points which are a fixed distance from
_two_points_ (the foci), i.e. that the sum of the two distances from any
point on the ellipse to the two foci is the same no matter where you are 
on the ellipse.

If you're dealing with a circle, the two foci points are just the same
point.  Let O be the center of the circle and let P be any point on the
ellipse.  As in your equation, I'll let PO be the distance between O and P.
So then we have PO = a, where a is the radius of the circle.  That means
that if any point P in the whole plane satisfies this equation (once we've
picked an a), it qualifies as a point on the circle.

Now let's rewrite this equation as PO + PO = 2a.  I've just multiplied 
by 2. Now if we let the circle have two centers instead of just one, i.e. 
replace the point O with your foci f and r, we'll get the equation 
Pf + Pr = 2a. That's the equation you have.  So in a sense, a is the 
"radius" of the ellipse.  

I hope this helps clear things up.  Write back if you have more questions!

-Ken "Dr." Math
Associated Topics:
High School Conic Sections/Circles
High School Definitions
High School Geometry

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