jmc

Let $A=(a,a^2-9a+25)$ be a point on the parabola $y=x^2-9x+25.$ Then the distance from $A$ to line $x-y-8=0$ is $$\begin{array}{rl}\frac{|a-(a^2-9a+25)-8|}{\sqrt{2}}& =\frac{|-a^2+10a-33|}{\sqrt{2}}\\ & =\frac{|a^2-10a+33|}{\sqrt{2}}\\ & =\frac{|(a-5{)}^2+8|}{\sqrt{2}}.\end{array}$$We see that $(a-5{)}^2+8$ is minimized when $a=5,$ and the minimum distance is $\frac{8}{\sqrt{2}}=\boxed{4\sqrt{2}}.$