Suppose $S$ is a surface parametrized by $f$ and its Gauss map is denoted by $N$. Define a map $f_t(u,v)=f(u,v)+tN(f(u,v))$. Define a focal point $q$ of $S$ as follows: if there is $t\neq 0$ such that $(u,v)$ is a critical point of $f_t$, then $q=f_t(u,v)$ is a focal point of $S$.
Now for a point $p\in S$, let its normal line be parametrized by $\alpha(s)=p+sN(p)$. The question asks to prove that $q$ is a focal point of $S$ if and only if $q=\alpha(\frac{1}{k})$, where $k$ is a principal curvature of $S$ at $p$.
My intuition is that $\frac{1}{k}$ gives the radius of the tangential sphere at $p$, so $q=\alpha(\frac{1}{k})$ is the center of the tangential sphere. Is this correct? Or is there any other formal way to show this? And how should I proceed from here? Any help would be appreciated!
I don't know what you mean by "the tangential sphere." Just write down what it means for $(u,v)$ to be a critical point of $f_t$ ($Df_t(u,v)(\mathbf X) = \mathbf 0$ for some nonzero $\mathbf X\in\Bbb R^2$) and use the definition of principal curvatures.