7. If 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 $k$ equals $\qquad$
The directrix of the hyperbola is given by $x= \pm \frac{4}{3}$, thus $\frac{1}{k}<1$ and $\frac{1}{\sqrt{1-\frac{1}{k}}}=\frac{4}{3} \Rightarrow \sqrt{1-\frac{1}{k}}=\frac{3}{4} \Rightarrow k=\frac{16}{7}$.