How to turn a function r(t) into a quadratic surface equation?

Question

I encountered a question where I have to match some given quadratic surfaces with some given functions r(t). I have the solution below but what I don't get is how they got the equations using the given functions r(t). I would appreciate any help. Thank you!

Answer 1

$\langle r(t) \rangle = \langle x(t),y(t),z(t) \rangle$

For example, looking at C, we have that $x(t) = 2cos(t), y(t)=3sin(t),z(t)=3-4cos(t)-9sin(t)$

You can observe that $z(t) = 3-2x(t)-3y(t)$ and therefore you can conclude that this is the same as $z=3-2x-3y \to 2x+3y+z=3$ which is a plane.

Answer 2

You have to eliminate $t$ to obtain a relation between $(x,y,z)$ which represents a surface. Your eliminations is correct. The curves

A: Spiral on a sphere of unit radius

D: A helix creeping on an elliptic cylinder

E: The generator on a unit radius cylinder