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Re: Hyperbola
Posted:
Feb 21, 1995 3:42 AM


>how to find the asymptotes of the >hyperbola (without advance knowledge of the equation, and other than simply >"eyeballing" the asymptote).
I am not sure what you mean by "without advance knowledge of the equation" but I hope you find this helpful.
A hyperbola can be as the set of all points in a plane the difference of whose distances from two fixed points (the foci) is a positive constant. Let (c,0) and (c,0) be the foci (this hyperbola will open to the sides and be symmetric about the yaxis). Let 2a be the positive constant. So if a point (x,y) is on the hyperbola, we know that the distance from (x,y) to (c,0) minus the distance from (x,y) to (c,0) is 2a. This statement can be transformed into the equation of the hyperbola that when simplified becomes (x^2/a^2)  (y^2/(c^2a^2)) = 1. Define b = sqrt(c^2a^2) and consider the rectangle with vertices (a,b), (a,b), (a,b), and (a, b). The diagonals of this rectangle are the asymptotes. The equations of these diagonals is y = +/ (b/a)x
So, if you know the foci((c,0) and (c,0)) and the common difference (2a) you can find the asymptotes.
If the hyperbola opens up and down (instead of side to side) the foci are (0,c) and (0,c). b is defined as before and the corners of the rectangle become (b,a), (b,a), (b,a) and (b,a) and the equations of the diagonals are y = +/ (a/b)x.
 Murphy Waggoner Department of Mathematics Simpson College 701 North C Street Indianola, IA 50125 waggoner@storm.simpson.edu 



