Derivation of parametric equations of a hyperbola

Question

Can somebody please show me how to derive the parametric coordinates of a hyperbola from $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ without just substituting them in?

Thanks

Answer 1

You could approach this as follows - first use some intuition to determine that the parametri equations should be of the form:$$x=a\cdot f(t)$$$$y=b\cdot g(t)$$as these eliminate $a$ and $b$ to given:$$(f(t))^2-(g(t))^2=1\tag{1}$$So now you just need to find functions $f(t)$ and $g(t)$ that satisfy (1).

One trig identity that comes to mind is:$$\sec^2(t)=1+\tan^2(t)$$which can be rearranged to give:$$\sec^2(t)-\tan^2(t)=1$$Thus one possible parametric solution would be:$$x=a\cdot sec(t)$$$$y=b\cdot \tan(t)$$