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Hyperbolic Functions

In the two Sketches below you can explore the Hyperbolic Sine and Cosine.
In the left Sketch both are expressed with respect to the variable t, while the second Sketch illustrates how they constitute hyperbolas parametrically.
The four Control Bars below affect both Sketches. The variable t in the left Sketch is dragable and the corresponding points on curves are shown in both Sketches.

Graphs of Hyperbolic Sine and Cosine
y = r cosh at
y = s sinh bt
The parametric curves in Phase Plane
ì
í
î
x
=
r cosh at
y
=
s sinh bt
        ì
í
î
x
=
r sinh at
y
=
s cosh bt
[GeoScript-File] [GeoStyle-File] [GeoScript-File] [GeoStyle-File]

[GeoScript-File]
Problems: 1. Keep r = a = s = 1. Find a value b so that the two hyperbolic curves on the right look like conjugates having the same straight line as their asymptote.
2. How do the parametric curve behave when the sign of r is changed negative?
3. Keep fixed a = s = b = 1 and see what happens when r goes to zero.
4. What if you keep r = s = 1 and change a and b so that they are equal?