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/