Suppose that $\sigma :[a,b]\to \mathbb{R}^3$ is a curve such that for all $t_0 \in [a,b]$ the line through $\sigma(t_0)$ and parallel to $\sigma'(t_0)$ passes through the same point $x_0 \in \mathbb{R}^3$.
Why $\sigma([a,b]) \subset l$ where $l$ is an affine line in $\mathbb{R}^3$?
I tried derivating $\sigma(t_0)-x_0+t(t_0)\frac{\sigma'(t_0)}{\|\sigma'(t_0)\|}$ where $t(t_0)$ is such that $\sigma(t_0)-x_0+t(t_0)\frac{\sigma'(t_0)}{\|\sigma'(t_0)\|}=0$.
HINT: Assume the point in question is the origin and assume the curve is arclength-parametrized. Write $\sigma(s) = \lambda(s)T(s)$, where $T=\sigma’$ is the unit tangent vector. (For all these sorts of problems, in general you want to set yourself up to apply the Frenet formulas.)