Arc length parameterization and collinearity

Question

I have $c:I\rightarrow\mathbb(R)^2$ an Arc length parameterized curve. $c$ stays inside a disk of radius $r$ and a $t_0, \Vert c(t_0) \Vert=r $

I need to show that $c''(t_0)$ is colinear to $c(t_0)$.

I tried differentiating $\Vert c(t) \Vert ^2$ but i get $2\langle c'(t),c(t)\rangle$ which isn't much help. We know $\Vert c'(t)\Vert =1$ so $\Vert c'(t)\Vert ^2=1$ so by differentiating i get $\langle c''(t),c'(t)\rangle =0$ but this isn't what i need and might not even be correct.

Any idea on how to get this done?

Answer 1

For the collinearity: From $c(t_0)=r$ we know that $\|c(t)\|^2$ has a maximum in $t_0$, and hence $0=(\|c(t_0)\|^2)'=2\langle c(t_0), c'(t_0)\rangle$ we conclude that $c(t_0)$ and $c'(t_0)$ are perpendicular. Furthermore, as $\langle c',c''\rangle=0$ we know that also $c''$ is perpendicular to $c'$. Hence $c(t_0)\parallel c''(t_0)$.