What is the slope of any line perpendicular to the line passing through (2,17) and (8,-2)?

The slope of the line passing through $\left(2 , 17\right)$ and $\left(8 , - 2\right)$ is
$L = \frac{- 2 - 17}{8 - 2} = - \frac{19}{6}$
Let $S$ the slope of the perpendicular line then we have that
$L \cdot S = - 1 \implies S = - \frac{1}{L} \implies S = \frac{6}{19}$