A tangent to the hyperbola $\mathrm{y}=\frac{\mathrm{x}+9}{\mathrm{x}+5}$ passing through the origin is

${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)}=\frac{-4}{{\left({\mathrm{x}}_1+5\right)}^2}$ equation of the tangent is $\mathrm{y}-{\mathrm{y}}_1=\frac{-4}{{\left({\mathrm{x}}_1+5\right)}^2}\left(\mathrm{x}-{\mathrm{x}}_1\right)$

The tangent is passing through origin then we get the point of contact and hence we get the tangent equation.