The range of g so that we have always a chord of contact of

 tangents drawn from a real point  $( a , a)$  to the circle  

$x^2+y^2+2gx+4y+2=0,$ is 

For a real chord of contact of tangents drawn from 

$(\alpha ,\alpha )$ to the circle $x^2+y^2+2gx+4y+2=0$ the point  $(\alpha ,\alpha )$

must lie outside the circle. 

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\alpha }^2+{\alpha }^2+2g\alpha +4\alpha +2>0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\alpha }^2+g\alpha +(2\alpha +1)>0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\alpha }^2+(g+2)\alpha +1>0\end{array}$

Since a assumes real values. Therefore, 

$\begin{array}{l}(g+2{)}^2-4>0\\ \Rightarrow g^2+4g>0\Rightarrow -4<g<0\Rightarrow g\in (-4,0).\end{array}$