Suppose I have a line segment $L$ in 3D: $$x=a_1(1-t)+b_1t$$ $$y=a_2(1-t)+b_2t$$ $$z=(a_1^2+a_2^2-k_1^2)(1-t)+(b_1^2+b_2^2-k_2^2)t$$
Because $L$ is line segment then $0\leq t\leq 1$.
And defining paraboloid $P$ in 3D: $$P:z=2x^2+2y^2-1$$
Where $a_1,a_2,b_1,b_2,k_1,k_2$ are all variables and $k_1$ and $k_2$ are positive numbers.
I want to put some constraints on these variables such that $L$ and $P$ intersect or do not intersect.
I know by substituting $L$ into $P$ and then solving for $t$ where $0\leq t\leq 1$ is the solution of the intersection. But still i could not figure out about the constraints i am looking for.
Thanks for any suggestions.