$y = mx + 6$ is a tangent to the hpyerbola
$ \frac{x^2}{100} - \frac{y^2}{49} = 1$ ...(i)
Tangent at $(x_1 , y_1)$ to the hyperbola is given
by $ \frac{ x x_1}{100} - \frac{y y_1}{49} = 1$ ...(ii)
Since equation $y = mx + 6$ and (ii) represent same line, these equations should be identical and hence
$\frac{\frac{x_{1}}{100}}{-m} = \frac{\frac{-y_{1}}{49}}{1} = \frac{1}{6} \Rightarrow x_{1} = - \frac{100\,m}{6} , y_{1} = - \frac{49}{6}$
$(x_1 , y_1)$ lies on the hyperbola, therefore satisfies its equation
$\frac{\left(-\frac{100m}{6}\right)^{2}}{100} - \frac{\left(-\frac{49}{6}\right)^{2}}{49} = 1 \Rightarrow m = \sqrt{\frac{17}{20}}$