The distance between the chords of contact of the tangent to the circle $x^2+y^2+2gx+2fy+c=0$

from the origin and the point $(g,f)$ is 

The equations of chords of contact of the tangents drawn from the origin and the point (g, f) to the given circle are respectively 

$gx+fy+c=0$          …(i)

and, $2gx+2fy+g^2+f^2+c=0$         …(iii)

Clearly, (i) and (ii) are parallel. Therefore, the distance 'd' between them is given by 

$d=\frac{g^2+f^2+c}{\sqrt{4g^2+4f^2}}-\frac{c}{\sqrt{8^2+f^2}}=\frac{g^2+f^2-c}{2g^2+f^2}$