Let \(f(x)=x^2+9, g(x)=\frac{x}{x-9}\) and \(a =f \circ g(10), b =g \circ f(3)\). If \(e\) and \(l\) denote the eccentri…

\(f\left(x\right)={x}^{2}+9,g\left(x\right)=\frac{x}{x-9}\)

\(a=f\left(g\left(10\right)\right)=f\left(\frac{10}{10-9}\right)\)

\(=f\left(10\right)=109\)

\(b=g\left(f\left(3\right)\right)=g\left(9+9\right)\)

\(=g\left(18\right)=\frac{18}{9}=2\)

\(E:\frac{{x}^{2}}{109}+\frac{{y}^{2}}{2}=1\)

\({\mathrm{e}}^{2}=1-\frac{2}{109}=\frac{107}{109}\)

\(ℓ=\frac{2(2)}{\sqrt{109}}=\frac{4}{\sqrt{109}}\)

\(8{\mathrm{e}}^{2}+{ℓ}^{2}=\frac{8(107)}{109}+\frac{16}{109}\)

\(=8\)