How to find the intersection of 2 3d parametric equations

Question

Parametric Equation

I have 2 parametric equation, $$(100\sin(t),100\cos(t),2t^2+200)$$ and, $$(100\cos(s), 100\sin(s), 2s^2+160)$$ How do I find the interception of these two parametric equations. I have tried a lot using simultaneous equations but it is really hard and a nearly unsolvable algebra equation. Thanks

Answer 1

Hint. Set corresponding coordinates equal. The last one gives $t^2+100=s^2+80,$ or $$t^2+20=s^2.$$ This tells us that $s^2\ge 20,$ or in other words that $|s|\ge 2\sqrt 5.$ The others give $\sin t=\cos s$ and $\cos t=\sin s,$ or in other words $\cos s-\cos(π/2-t)=0$ and $\cos t-\cos(π/2-s)=0,$ or $$-2\sin(s-t+π/2)/2\sin(s+t-π/2)/2=0$$ and $$-2\sin(t-s+π/2)/2\sin(t+s-π/2)/2=0.$$ Can you now continue?