Ellipse, ellipse and ellipse :P

Geometry Level 4

Consider the ellipse \(\frac{x^2}{f(k^2+2k+5)}+\frac{y^2}{f(k+11)}=1\), where \(f(x)\) is a positive decreasing function, then the value of \(k\) for which major axis coincides with the \(x-axis\) can be expressed in the form of \(k \in (a,b)\). Find \(a+b\)

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