Let \(\frac{{x}^{2}}{f\left({a}^{2}+7a+3\right)}+\frac{{y}^{2}}{f(3a+15)}=1\) represent an ellipse with major axis along…

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

[JEE Main 2026, 8 Apr (Shift 2)]