To prove that a vector $x(t)$ lies in a plane. Prove that vector $x(t)=t\,\hat{i}+\left(\dfrac{1+t}{t}\right)\hat{ j}+\left(\dfrac{1-t^2}{t})\right)\hat{k}$ lies in a curve.
   I am puzzled. Don't know how to approach it.

The vectors lie in a plane, because the functions $\{f=t,g=(1+t)/t,h=(1-t^2)/t\}$ are not linearly independent. Namely, $f-g+h=-1$.
It is obvious that they also lie in a curve: there is a single real parameter to be varied!
