Show the motion of a particle occurs along a quadric surface

Question

Given the formula $g(t)=(t \sin \theta,t\cos \theta, \sqrt{3}t),$ $0\leq t\leq4\pi$, how can I show this motion to be along a quadric surface?

Answer 1

Your particle is travelling on an "elliptic cone". To see this, note that for any point (note that we lose $t$ dependence in our computation) $$ (x,y,z)=(t\sin \theta,t\cos \theta,\sqrt{3}t) $$ defined by your parametric curve, you have $$ 3x^2+3y^2-z^2=0\implies z^2=3(x^2+y^2) $$