Minimum distance between a surface and a point I created this problem which is inspired by some problems I encountered in this site.
What is the minimum distance between point $(1,1,1)$ and a surface defined by $z-xy=10$?
The answer is $\sqrt{18}$ but I am curious how others solve this kind of problem.

We want to minimize 
$$\sqrt{(x-1)^2+(y-1)^2+(xy+10-1)^2}$$
which can be written as
$$\begin{align}&\sqrt{x^2-2x+1+y^2-2y+1+x^2y^2+18xy+81}
\\\\&=\sqrt{(y^2+1)x^2+(18y-2)x+y^2-2y+83}
\\\\&=\sqrt{(y^2+1)\bigg(x^2+\frac{18y-2}{y^2+1}x\bigg)+y^2-2y+83}
\\\\&=\sqrt{(y^2+1)\bigg(\bigg(x+\frac{9y-1}{y^2+1}\bigg)^2-\bigg(\frac{9y-1}{y^2+1}\bigg)^2\bigg)+y^2-2y+83}
\\\\&=\sqrt{(y^2+1)\bigg(x+\frac{9y-1}{y^2+1}\bigg)^2-\frac{(9y-1)^2}{y^2+1}+y^2-2y+83}
\\\\&=\sqrt{(y^2+1)\bigg(x+\frac{9y-1}{y^2+1}\bigg)^2+\frac{(y^2-y-8)^2}{y^2+1}+18}\end{align}$$
So, the minimum distance is $\color{red}{3\sqrt{2}}$ which is attained when
$$x+\frac{9y-1}{y^2+1}=y^2-y-8=0\quad \text{and}\quad z=xy+10$$
i.e.
$$(x,y)=\bigg(\frac{1\pm\sqrt{33}}{2},\frac{1\mp\sqrt{33}}{2},2\bigg)$$
