Limit, Continuity and Differentiability: JEE Main 2022 Mathematics question

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 ________________.