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/
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