So this problem is really confusing me.
Here is a hint we had: (Hint: since the Frenet frame is a frame, write $x = aT + bN + cB$ and work from there)
I had an initial idea based on the hint, but my teacher suggested a different method. He said he said he took derivatives of $(X\cdot X) $ and then used the frenet equations to solve for it.
This is something we worked through together and I completely understand these steps.
\begin{align} x\cdot x &= \text{constant}\\ \Rightarrow x\cdot x' &= 0 \\ \Rightarrow x\cdot T &= 0\\ (x.T)' &= x'\cdot T + x\cdot T' \\ &= T\cdot T + x\cdot (\kappa N) \\ &= 1 + \kappa x\cdot N = 0\\ \Rightarrow x\cdot N = -1/\kappa \end{align} and then take more from there.
My question, is where do I go from there. In working this out we have something that equals $-1/\kappa$ (and the initial problem has $1/\kappa$). The problem also has a $t/\kappa$ which could just be $t$ times $1/\kappa$. So, would I do similar steps to get something in the form of $1/\kappa$ and then have what I need to solve the problem? Like we have done no examples of anything like this in my class, so I do not know what to do next.