Consider the following problem (Exercise 2.55 in Montiel and Ros's *Curves and Surfaces*, 2nd Edition):

Let $S = \{p \in \mathbb{R}^3 \ | \ |p|^2 - \langle p, a \rangle^2 = r^2\}$, with $|a|=1$ and $r>0$, be a right cylinder of radius $r$ whose axis is the line passing through the origin with direction $a$. Prove that $T_pS = \{v \in \mathbb{R}^3 \ | \ \langle p, v \rangle - \langle p, a \rangle \langle a, v \rangle = 0\}$. Conclude that all the normal lines of $S$ cut the axis perpendicularly.

Now, $S = f^{-1}(r^2)$ where $f(p) = |p|^2 - \langle p, a \rangle^2$. But $$ (df)_p[v] = 2 \langle p, v \rangle - 2 \langle p, a \rangle \langle a, v \rangle, $$ hence the first part.

Now, How to show the second part?

Any hints will be the most appreciated.