The slope of the tangent to a curve \(C:y = y(x)\) at any point \(( x,y)\) on it is \(\frac{2e^{2x} - 6e^{- x} + 9}{2 + …

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

[JEE Main 2022, 25 Jul (Shift 1)]