<s>[INST] The ellipse of the maximum area contained between the curves $\frac{\pm 1}{x^2+c} $ 
The ellipse of the maximum area contained between the curves $\frac{\pm 1}{x^2+c}, c > 0 $       

It seems to me that the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ of the maximum area when $ a = c; b = \frac{1}{c} $
That is, it looks like this:

The problem is that I'm not sure that an ellipse here can be a finite area
I will be glad to any hint or solution
 [/INST] Expanding in Taylor series around $x=0$
$$
y = b\sqrt{1-\left(\frac{x}{a}\right)^2} = b-\frac{b x^2}{2 a^2}-\frac{b x^4}{8 a^4}+O\left(x^5\right)
$$
and
$$
y = \frac{1}{(x^2+c)} = \frac{1}{c}-\frac{x^2}{c^2}+\frac{x^4}{c^3}+O\left(x^5\right)
$$
Now solving
$$
\left\{
\begin{array}{rcl}
b & = & \displaystyle\frac{1}{c}\\
\displaystyle\frac{b}{2 a^2} & = & \displaystyle\frac{1}{c^2}
\end{array}
\right.
$$
we get 
$$
a = \sqrt{\frac{c}{2}}, b = \frac{1}{c}
$$
and the area is 
$$
S = \pi a b = \frac{\pi}{\sqrt{2 c}{}}
$$
NOTE
In the series expansion the third term is negative for the ellipse.This guarantees the inclusion.
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