Question:
Find a parameterization of the paraboloid $900z = 25x^2 + 36y^2$.
My Work
$25x^2 + 36y^2 = 900z$
$\implies (5x)^2 + (6y)^2 = (30\sqrt{z})^2$
Can we can represent this equation using cylindrical coordinates?
$(x,y,z) = (\rho \cos(\theta), \rho \sin(\theta), \zeta)$ where $\rho = $ radius and $2\pi \ge \theta \ge 0$.
I think I'm on the right track here, but I've spent hours unsuccessfully pondering over this problem and doing research. I want to get it into cylindrical coordinates and then parameterise it in terms of $u$ and $v$, but I'm just completely stuck.
I would greatly appreciate it if people could please take the time to show me the correct reasoning and solution for this problem.
Best Answer
Whenever you have an equation of the form $z = f(x,y)$, you can use $x$ and $y$ as parameters. So one possible set of parametric equations is $$ x = u \quad ; \quad y = v \quad ; \quad z = \frac{1}{900}(25u^2 + 36v^2) $$ If you really want to use trig functions, then: $$ x = \tfrac15 r \cos \theta \quad ; \quad y = \tfrac16 r \sin \theta \quad ; \quad z = \tfrac{1}{900}r^2 $$ True polar coordinates are going to be messy. The problem is that each curve of the form $z = \text{constant}$ is an ellipse, and the equation of an ellipse in polar coordinates is a bit complicated.