Re: Conic sections

Ed Murphy wrote:
On Thu, 03 Aug 2006 00:29:43 GMT, Mark P
<usenet@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

This is a follow-up of sorts to the recent "Horsing Around" puzzle.

Most of us learn in a school that an ellipse is the set of all points who summed distance from two foci is constant and that a hyperbola is the set of all points who "differenced" distance from two foci is constant.

But I hadn't known until the above mentioned puzzle that the set of all points whose divided distance from two foci is constant defines a circle. (And based on an informal survey of a few coworkers, this seems not to be common knowledge.)

Anyway, to the question... is there any intuitive or geometric way to see the preceding result?

This only became intuitive after I went through the formal
derivation, but:

The equality describing the locus has something resembling a
definition of a circle [ (x-a)^2 + (y-b)^2 ] on both sides. We
can subtract one from the other, then complete the square to get
something that is precisely the definition of a circle.

Okay, formal derivation follows.


[snip]

No complaints with your derivation and I followed the same process when I first found that it was a circle, but I was hoping for something more intuitive. You know the sort of argument: "If we draw this triangle and that triangle then you can see that they're similar yada yada..." (It's easy when it's made up :)) Basically I'm looking for something other than the straightforward algebraic derivation.
.

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