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Parabola Focal Point

Date: 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
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Associated Topics:
High School Equations, Graphs, Translations

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