The equation of the tangents drawn from the origin to the circle

$x^2+y^2+2gx+2fy+f^2=0$ is

Equation to the pair of tangents drawn from $(0,0)\text{ to }{x}^2+y^2+2gx+2fy+f=0$ is

${\left(gx+fy+f^2\right)}^2=f^2\left(x^2+y^2+2gx+2fy+f^2\right)$

$\Rightarrow g^2{x}^2+2fgxy+f^2{y}^2+2g{f}^{2}x+2f^{3}y+f^4=f^2{x}^2+f^2{y}^2+2g{f}^{2}x+2f^{3}y+f^2$

$\Rightarrow \left(f^2-g^2\right)x^2-2fgxy=0\Rightarrow x\left[\left(f^2-g^2\right)x-2fgy\right]=0$

$\therefore$The equation of the tangents are $x=0,\left(f^2-g^2\right)x-2fgy=0$