For the hyperbola $\frac{{\mathrm{x}}^2}{{\cos }^2 \alpha }-\frac{{\mathrm{y}}^2}{{\sin }^2 \alpha }=1$ which of the following remains constant when $\mathrm{\alpha }$ varies?

$$\begin{gathered} \frac{{\mathrm{x}}^2}{{\cos }^2\mathrm{\alpha }}-\frac{{\mathrm{y}}^2}{{\sin }^2\mathrm{\alpha }}=1 \\ {\mathrm{a}}^2={\cos }^2\mathrm{\alpha }, {\mathrm{b}}^2={\sin }^2\mathrm{\alpha } \\ \mathrm{Now} e=\sqrt{\frac{a^2+b^2}{a^2}} \\ \Rightarrow \mathrm{e}=\sqrt{\frac{{\cos }^2\mathrm{\alpha }+{\sin }^2\mathrm{\alpha }}{{\cos }^2\mathrm{\alpha }}} \\ \Rightarrow \mathrm{e} =\frac{1}{\mathrm{cos\alpha }} \\ \mathrm{Now} ae=\mathrm{cos\alpha }\left(\frac{1}{\mathrm{cos\alpha }}\right) \\ =1 \end{gathered}$$

$\therefore$ abscissae of foci is constant when $\text{'}\mathrm{\alpha }\text{'}$ varies.