```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 theexistence of conics other than parabolae!              John Conway
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