Value of the coordinates in parametric form of hyperbola

Question

We know a hyperbola can be expressed in the form of$$ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$ where $(h,k)$ is it's center. I've learnt that in the parametric form, we take $$x= h + a\sec t$$ and $$ y = k + b\tan t $$ These values satisfy the given equation. But so does $$x= h + a\csc t$$ and $$y=k+b\cot t$$ Then why aren't these second values of $x$ and $y$ taken as parameters? Does this cause a difference in the graphs?

Answer 1

The answer is largely convention and convenience.

Your alternative parameterization will work, but since we often parameterize curves using $x=\cos{\theta}$ and $y=\sin{\theta}$ for internal consistency with the geometric definitions for $\sin$ and $\cos$ it is a more natural choice to use $\sec$ for the $x$ value if $\cos$ won't work.

From there, because we need to make $x^{2}-y^{2}=1$ the identity of choice is $\sec^{2}{\theta}-\tan^{2}{\theta}=1$ and the parameterization then follows.