Bài giải:

$C=\sqrt{{{x}^{2}}+{{y}^{2}}-2xy+2x-2y+10}+{{y}^{2}}-4y+200=\sqrt{{{(x-y+1)}^{2}}+9}+{{(y-2)}^{2}}+196$

$\Rightarrow C\ge \sqrt{9}+196=3+196=199$

Dấu “=” xảy ra  $\Leftrightarrow \left\{ \begin{align}& x-y+1=0 \\& y-2=0 \\\end{align} \right.$

$\Leftrightarrow \left\{ \begin{align}& x-2+1=0 \\& y=2 \\\end{align} \right.$

$\Leftrightarrow x=1;y=2$. Vậy min C = 199