5 The conic section $\frac{(x-1)^{2}}{4}+\frac{(y-2)^{2}}{a}=1$ has a directrix with the equation $x=9$, then the value of $a$ is
$5 \frac{15}{4}$. Since the equation of the curve is
$$
\frac{(x-1)^{2}}{4}+\frac{(y-2)^{2}}{a}=1 \text {, }
$$

the equation of the directrix is
$$
x-1=\frac{4}{\sqrt{4-a}},
$$

and by the problem statement,
$$
9-1=\frac{4}{\sqrt{4-a}},
$$

solving for $a$ yields $a=\frac{15}{4}$.