If the line y=m x+7

Since, $1 \mathrm{x}+\mathrm{my}+\mathrm{n}=0$ is a normal to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$,

then $\frac{a^{2}}{l^{2}}-\frac{b^{2}}{m^{2}}=\frac{\left(a^{2}+b^{2}\right)^{2}}{n^{2}}$

but it is given that $m x-y+7 \sqrt{3}$ is normal to hyperbola

$\frac{x^{2}}{24}-\frac{y^{2}}{18}=1$

then $\frac{24}{m^{2}}-\frac{18}{(-1)^{2}}=\frac{(24+18)^{2}}{(7 \sqrt{3})^{2}} \Rightarrow m=\frac{2}{\sqrt{5}}$