If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are two points on the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$  and the chord joining these two points passes through the focus $(\mathrm{ae}, 0)$ then $\mathrm{e} \cos \frac{\mathrm{\alpha }-\mathrm{\beta }}{2}=$

$$\begin{gathered} \mathrm{Given} \mathrm{hyperbola} \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1 \\ \mathrm{Let} \mathrm{P}\left(\mathrm{\alpha }\right)=(\mathrm{a} \sec \mathrm{\alpha }, \mathrm{b} \tan \mathrm{\alpha }) and \mathrm{Q}\left(\mathrm{\beta }\right)=(\mathrm{a} \sec \mathrm{\beta }, \mathrm{b} \tan \mathrm{\beta }) be two points on given hyperbola \\ \mathrm{Now} \mathrm{equation} \mathrm{of} \mathrm{chord} \mathrm{joining} \text{'}\mathrm{P}\text{'} \mathrm{and} \text{'}\mathrm{Q}\text{'} \mathrm{is} \frac{\mathrm{x}}{\mathrm{a}}\cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)-\frac{\mathrm{y}}{\mathrm{b}}\sin \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)=\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right) \\ \mathrm{It} \mathrm{passes} \mathrm{through} \mathrm{focus} (\mathrm{ae}, 0) \mathrm{then} \\ \Rightarrow \mathrm{e} \cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)=\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right) \end{gathered}$$