1. The ellipse $x^{2}+k y^{2}=1$ and the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{5}=1$ have the same directrix. Then $k=$ $\qquad$ .
$=1 \cdot \frac{16}{7}$.
The equations of the directrices of the hyperbola are
$$
x= \pm \frac{4}{\sqrt{4+5}}= \pm \frac{4}{3} \text {, }
$$

The equations of the directrices of the ellipse are $x= \pm \frac{1}{\sqrt{1-\frac{1}{k}}}$. Then $\frac{1}{\sqrt{1-\frac{1}{k}}}=\frac{4}{3} \Rightarrow 1-\frac{1}{k}=\frac{9}{16} \Rightarrow k=\frac{16}{7}$.