Application Of Derivatives

Let the function \(f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0\), be decreasing in \((0, \mathrm{a})\) and increasing in \((\mathrm{a}, 4)\). A tangent to the parabola \(y^{2}=4 a x\) at a point \(\mathrm{P}\) on it passes through the point \((8 \mathrm{a}, 8 \mathrm{a}-1)\) but does not pass through the point \(\left(-\frac{1}{a}, 0\right)\). If the equation of the normal at \(P\) is : \(\frac{x}{\alpha}+\frac{y}{\beta}=1\), then \(\alpha+\beta\) is equal to ________________.