Parametric curve: $x=\frac{a}{2}(t+\frac{1}{t})$, $y=\frac{b}{2}(t-\frac{1}{t})$?

Question

What kind of shape is the parametric curve described by:

$$x=\frac{a}{2}(t+\frac{1}{t})$$ $$y=\frac{b}{2}(t-\frac{1}{t})$$

$a,b \in\mathbb{R^+}$

?

Answer 1

A set of hyperbolas with x-, y- axes as symmetry axes.

Although not asked for, the above can be seen as projections of a surface for each of two parameters, one extra parameter can be brought in.

To connect further to paraboloids, a 3D parametrization is necessary with $two$ parameters.

$$ x=\dfrac{u}{2}(t+\frac{1}{t}) ; y=\dfrac{u}{2}(t-\frac{1}{t}) ; z = u ;$$

Answer 2

Hint: Let $t=e^u$ for $t>0$, and $t=-e^u$ for $t<0$, then use $\cosh^2u-\sinh^2u=1$.