jmc

To get the equation of the asymptotes, we replace the $1$ on the right-hand side with $0,$ giving the equation$$\frac{x^2}{100}-\frac{y^2}{64}=0.$$(Notice that there are no points $(x,y)$ which satisfy both this equation and the given equation, so as expected, the hyperbola never intersects its asymptotes.) This is equivalent to $\frac{x^2}{100}=\frac{y^2}{64},$ or $\frac{y}{8}=\pm \frac{x}{10}.$ Thus, $y=\pm \frac{4}{5}x,$ so $m=\boxed{\frac{4}{5}}.$