[alogo] 1. Hyperbola Asymptotics

Look at Hyperbola.html for the definition of the hyperbola and its equation with respect to its principal axes:


For c = a+b, c = |FF*|/2 being half the focal-distance and e = c/a, which is the eccentricity of the hyperbola (e>1), the angle of its asymptote with the x-axis is given by:

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Referring the curve to coordinates (u, v), with respect to its asymptotic lines and calculating the relations between the two kinds of coordinates, we find that

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The figure shows also the inverse path. How to determine the axes of the hyperbola, knowing its equation uv=c/4, with respect to its asymptotic lines: Draw the circle with radius c, find ABCD, F, F* and E. This procedure, of the definition of the hyperbola, from two intersecting lines (playing the role of asymptotics) and positive number c is realized dynamically with EucliDraw in the file HyperbolaWRAsymptotics.html .

[alogo] 2. A property of asymptotics

For every point P on the hyperbola draw a parallel to the minor axis intersecting the asymptotes at {I,J}. Then
PI*PJ = b2.

Follows from the last basic relation.


Remark This property generalizes in the following way. Take an arbitrary secant through P intersecting the asymptotes at {I',J'} then

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In particular, if the secant I'J' moves parallel to itself then the product PI'*PJ' remains constant. Thus, if P comes to the position of P' the product has the same value and this implies that PI'=P'J'.

[alogo] 3. Construction problem

To construct an hyperbola from its asymptotes and a point P on the curve.

Apply the previous equation to find b. Then construct rectangle ABCD determining the various basic elements (focus etc.) of the hyperbola.

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