>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 y-axis). 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^2-a^2)) = 1. Define b = sqrt(c^2-a^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.
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