Associated Topics || Dr. Math Home || Search Dr. Math

Meaning of Value of b in Hyperbola Equation

```Date: 05/06/2007 at 21:01:54
From: Don
Subject: justification for hyperbola formula

I'm teaching conic sections, and I have been unable to find a
justification for why in a hyperbola does a^2 + b^2 = c^2.  You can
easily justify a^2 = b^2 + c^2 in an ellipse by looking at special
points.  But I have yet to find a comparable explanation for
hyperbolas.  Textbooks just give you the formula and never explain
where it comes from.

```

```

Date: 05/07/2007 at 08:35:45
From: Doctor Fenton
Subject: Re: justification for hyperbola formula

Hi Don,

Thanks for writing to Dr. Math.  The relationship is true because
that is the DEFINITION of b.  b doesn't correspond to any geometric
feature in the specification of the hyperbola, if you are using the
description as the points whose distances to the two foci differ by a
given amount.  The foci are determined by the number c, and the given
difference determines the coordinates of the vertices a, and with
these two numbers, you can derive the equation

x^2      y^2
--- - --------- = 1
a^2   c^2 - a^2

(for a hyperbola centered at the origin, with foci (+/-c,0) and
vertices (+/-a,0)).  Since c^2 > a^2, c^2 - a^2 > 0, there is a
positive number b such that b^2 = c^2 - a^2, and using this clearly
simplifies the denominator of y^2 in the formula above.  The point is
that a and c are enough information to completely determine the
hyperbola.  No value of b is needed, and it is simply introduced to
simplify the notation.

Actually, the ellipse is similar: the foci and vertices (or the sum of
the distances to the foci, which determines the vertices) are all that
is needed to define the ellipse.  It turns out that if you introduce
the semi-minor axis b, you simplify the equation, and the quantity has
a geometric meaning, but this was not part of the original
specification.

The hyperbola also has asymptotes

y   x             y   x
- - - = 0    and  - + - = 0
b   a             b   a

and these can be found by drawing a box with corners (+/-a,+/-b), but
that box is not part of the original specification of the hyperbola.
It is something you find after you have found the hyperbola.

In both cases (ellipse and hyperbola), the definition essentially
specifies a and c, and b is introduced for convenience.  However, once
you deduce the geometric significance of b, it offers an alternative
way of specifying the conic.  Any two of a, b, and c can be given and
the third quantity determined.

Does that help?

- Doctor Fenton, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Conic Sections/Circles
High School Conic Sections/Circles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search