For c² = a²+b², 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:

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

The figure shows also the inverse path. How to determine the axes of the hyperbola, knowing its equation uv=c²/4, 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 c² is realized dynamically with EucliDraw in the file HyperbolaWRAsymptotics.html .

PI*PJ = b

Follows from the last basic relation.

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'.

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

HyperbolaProperty.html

HyperbolaWRAsymptotics.html

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