The slope of the tangent to a curve $C: y=y(x)$ at any point $(x, y)$ on it is $\dfrac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}$. If $C$ passes through the points $\left(0, \tfrac{1}{2}+\tfrac{\pi}{2\sqrt{2}}\right)$ and $\left(\alpha, \tfrac{1}{2}e^{2\alpha}\right)$, then $e^{\alpha}$ is equal to :