Date: Nov 17, 1997 8:53 PM
Author: Peter Ash
Subject: Re: parabola through 4 points
Michael,

Thanks for your response. I don't fully understand your argument,

though.

Michael Keyton wrote:

>

> Peter has asked about finding the foci of a hyperbola knowing 5 points

> that lie on the hyperbola. I can not fully solve this, but I can find the

> center of the hyperbola. For the ellipse, I can find both the center and

> the foci. I think some extension about this construction should work for

> the hyperbola, but it is not yielding itself to me quickly.

> Ellipse: Construct any chord of the ellipse, construct a parallel chord.

> The line through the midpoints of two parallel chords passes through the

> center. (I do not know a proof of this without adding a point at infinity

> and looking at poles. I can give a reasonable argument using symmetry of

> the ellipse, but it lacks the proper rigor.) Repeat for a second chord,

> the interesection of the two lines is the center of the ellipse. This also

> works for finding the center of teh hyperbola.

I agree that the line through the midpoints of two parallel chords of

any

conic passes through the center of the conic. (Proof: The projective

transformation

that fixes the center of the conic and maps it to a circle takes

segments to segments, parallel lines to parallel lines, and midpoints of

segments to midpoints of segments.) But given five arbitary points on a

conic, how do you get two parallel chords? Am I missing something

obvious?

>

> Construct any circle centered at this point that intersects the ellipse in

> 4 points (use the five points, 3 can't be the vertices of the minor axis.)

> The perpendicular bisectors of the consecutive pairs of points are the

> major and minor axis. (In the hyperbola case, this will find the axis for

> the foci) Use compass and straightedge, knowing the relationships between

> the distance between the foci and the center and the semi-major and minor

> axis lengths fill find the foci length. This can be constructed easily.

How do you locate the points of intersection between a circle centered

at

the center of the conic and the conic? And then how do you find the

intersections of the axes with the ellipse? Once you've got that, of

course you are right, its easy to find the foci.

>

> Unfortunately, for the hyperbola where is the length of the minor axis?

>

It would seem you would have to know the asymptotes.

> Michael Keyton

> St. Mark's School of Texas

>