Date: Nov 11, 1997 3:05 PM
Author: John Conway
Subject: Re: construction of triangle of given perimeter, given point and angle

On Tue, 11 Nov 1997, Eileen M. Klimick Schoaff wrote:

> A parabola, on the other hand, is determined by any 4 of its points.
> John Conway>
> Am I missing something here? Don't 3 points determine a parabola if the axis
> of symmetry is either vertical or horizontal?

Yes they do. But not if it isn't. You did miss something!

But if we consider any axes,
> then there are an infinite number of parabolas passing through three points.

Yes, this is true. But I spoke of 4 points, not 3.

> The generic equation is ax^2 + bxy +cy^2 + dx + ey + f = 0.

This is the general conic, which is usually an ellipse or hyperbola
rather than a hyperbola.

If Jon Roberts is
> considering parabolas of the form y = ax^2 + bx + c, then knowing 3 points
> gives you three equations with three unknowns which can easily be solved --
> unless there is no solution.
> In the April 1997 issue of the Mathematics Teacher, a colleague of mine, Dr.
> Ellie Johnson, wrote an article "A Look at Parabolas with a Graphing
> Calculator". In this article she using the calculator to generate many
> solutions to the generic equation. Of course this just shows that given three
> points and restricting yourself to a parabola of the form y = ax^2 + bx + c,
> you can derive the equation. That does not, of course, construct it.
> Does the fourth point determine whether the axis of symmetry is vertical,
> horizontal, or rotated?

Yes, roughly speaking.

In ax^2 + bxy +cy^2 + dx + ey + f = 0, it looks like
> you need more than 4 points to determine a, b, c, d, e, f.

You do indeed need 5 points to determine the general conic.

> Then again, I am only a math education person and do not have a PhD in math so
> I am probably far in the dark.
> Eileen Schoaff
> Buffalo State College

I think as a math educator you really SHOULD have known of the
existence of conics other than parabolae!

John Conway