We set $\theta = \operatorname { arcosh } \left( \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } } \right) > 0$ and $\alpha = e ^ { - \theta }$. Show that $\alpha$ is a root of the polynomial
$$X ^ { 2 } - 2 \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } } X + 1$$
and deduce the expression of $\alpha$ in terms of the quantity $\beta = \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } }$.