Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant. 
Let $r(s)$ be a curve parametrized by arc length, and $\kappa,\kappa',\tau$ are non-zero. Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant. 

The teacher gave the hint "center = $r+(1/\kappa)N+(1/\kappa)'(1/\tau)B$". I know how to get the proof from this, but how can we show that this is the center? Please help. Thanks.

Big Hint: The standard technique in all such differential geometry problems is to write
$$ r = \lambda T + \mu N + \nu B$$
for some functions $\lambda$, $\mu$, and $\nu$. Differentiate, use what you're given, and use the Frenet equations and use the fact that $T,N,B$ form a basis to get three equations.
