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### Parabola Focal Point

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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.)?

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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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Associated Topics:
High School Equations, Graphs, Translations

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