Ellipoint !

If the above line touches the ellipse in a single point, then we have:

$\frac{x^2}{5} + \frac{(2x+a)^{2}}{9} = 1;$

or $9x^2 + 5(2x+a)^{2} = 45;$

or $29x^2 + 20ax + (5a^2 - 45) = 0;$

or $x = \frac{-20a \pm \sqrt{400a^2 - 4(29)(5a^2 - 45)}}{58};$

or $x = \frac{-20a \pm \sqrt{5220 - 180a^2}}{58}.$

In order to have a single point solution for $x$ , we require the discriminant to equal zero: $5220 - 180a^2 = 0 \Rightarrow a^2 = 29.$ Thus, $a^2 - 100 = \boxed{-71}.$