Definition of an EllipseDate: 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 |
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