Centres and radii of the given circles $x^{2}+y^{2}=9$
and $x^{2}+y^{2}+2 a x+2 y+1=0$ is $C_{1}(0,0), r_{1}=3$
and $C_{2}(-\alpha, 1)$ and $r_{2}=\sqrt{\alpha^{2}+1-1}=|\alpha|$
Since, two circles touch internally,
$\therefore C_{1} C_{2}=r_{1}-r_{2}$
$\Rightarrow \,\,\,,\sqrt{\alpha^{2}+1^{2}}=3-|\alpha|$
$\Rightarrow \,\,\,\alpha^{2}+1=9+\alpha^{2}-6|\alpha|$
$\Rightarrow \,\,\,\,6|\alpha|=8 $
$\Rightarrow |\alpha|=\frac{4}{3}$
$\Rightarrow \,\,\,\alpha=\pm \frac{4}{3}$