Parabola Focal PointDate: 7/3/96 at 17:55:57 From: Mr. Gil Kaelin Jr. Subject: Parabola Focal Point What are the rules for the focal point of a parabola? Is there a rule or rules in parabola dimension (i.e. is H:W a 1:1, must a parabola be circular, what arc is required, etc.)? I would like to understand this better. Please help. Date: 7/8/96 at 9:23:40 From: Doctor Brian Subject: Re: Parabola Focal Point Well, here's a little bit about parabolas, but not the definitive treatise on the subject: There are two equivalent ways of looking at them (a little algebra will turn one method into the other): 1. The graph of a second-degree polynomial function in one variable (known as the old y = ax^2 + bx + c rule usually seen in first or second year algebra class). 2. The set of all points that at the same time are equidistant from a given point and a given line (the point is the focus, and the line is the directrix). Now, if you know the vertex of the parabola, it's got to be exactly midway between the focal point and the directrix line (equidistant). One of the interesting things about a focal point is that the distance across the parabola through the focus is equal to four times the distance from the focus to the vertex. This isn't too tough to show, those side points on the parabola must be twice that distance each to the directrix, and therefore, to the focus. The arc isn't really circular. It's more like an infinite valley or mountain (depending on whether its vertex, or "turning point" is at the top or bottom). The steepness of the curve gets more steep the further away from the vertex, near which the curve *looks* more round before starting to straighten out....not that it actually straightens out, but it is less obviously curved further away from the vertex. -Doctor Brian, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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