The stated problem is :
Assume that $α$ is a regular curve in $R^2$ and all the normal lines of the curve pass though the origin. Prove that $α$ is contained in a circle around the origin. (Recall the normal line at $α(t)$ is the line through $α(t)$ pointing in the direction of the normal vector $N(t)$.
There is a hint provided:
If all normal lines pass through a fixed point, then for all $s \in I$, there is a $\lambda = \lambda(s)$ such that $$\alpha(s) + \lambda(s)n(s) = p $$ where $p$ is our supposed fixed point and we are suppose to take the derivative of this.
This is all fine, but there is a small technical point that is making me uncomfortable. How do we know $\lambda(s)$ is even continuous or differentiable in the first place?
Best Answer
Look at specific scalar components. Presumably, you know $n(s)$ is differentiable/continuous and never zero. Hence, at each $s$, there is a component $i$ such that $n_i(s)\neq0$, so by continuity this also holds in a small neighborhood of $s$. Now in this neighborhood, we have the scalar equation $\lambda(s)=\frac{p_i-\alpha_i(s)}{n_i(s)}$, and continuity/differentiability follows.
This should also show that if you strengthen the hypothesis to $\alpha,n$ being $C^k$, then so does your $\lambda$.
(I think this might also follow from some version of implicit function theorem.)