Intersection of two surfaces, $z=2x^2-y^2$ and $z=4$

Question

So the usual way to do this kind of thing is to first plug in $z=4$ to get $$4=2x^2-y^2$$ and now just use one of the variables as a parameter. So now i have a choice, should i use $x$ or $y$ as a parameter for this curve. Now depending on my choice the vector function of the curve will be different. Is the only difference between these function is the direction, in which the curve goes? One is say clockwise and the other one is counter-clockwise?

Thanks

Answer 1

This is just the hyperbola $\;\dfrac{x^2}2-\dfrac{y^2}4=1$ in the plane $z=4$. The asymptotes have equation $$2x^2-y^2=(\sqrt2x-y)(\sqrt2x+y)=0.$$