Let x^2/(f( a^2+7 a +3))+y^2/(f(3 a +15))=1 represent an ellipse with major axis along y -axis, where f is a strictly decreasing positive function on…

Let $\displaystyle \frac{x^2}{f\left(\mathrm{a}^2+7 \mathrm{a}+3\right)}+\frac{y^2}{f(3 \mathrm{a}+15)}=1$ represent an ellipse with major axis along $\displaystyle y$-axis, where $\displaystyle f$ is a strictly decreasing positive function on $\displaystyle \mathbf{R}$. If the set of all possible values of $\displaystyle a$ is $\displaystyle \mathbf{R}-[\alpha, \beta]$, then $\displaystyle \alpha^2+\beta^2$ is equal to :