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

$\begin{array}{r}y=\frac{x+9}{x+5}=1+\frac{4}{x+5}\\ \frac{dy}{dx}\text{ at }\left(x_1,y_1\right)=\frac{-4}{{\left(x_1+5\right)}^2}\end{array}$

$\therefore \text{ Equation of tangent is }$

$\begin{array}{r}y-y_1=\frac{-4}{{\left(x_1+5\right)}^2}\left(x-x_1\right)\\ y-1-\frac{4}{x_1+5}=\frac{-4}{{\left(x_1+5\right)}^2}\cdot \left(x-x_1\right)\end{array}$

$\text{ Since it passes through }(0,0)\text{. }$

${\left(x_1+5\right)}^2+4\left(x_1+5\right)+4x_1=0$

$x_1=-15\text{ or }{x}_1=-3\text{. So equations are }x+25y=0\text{ or }x+y=0\text{. }$