Let equation of ellipse is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$\therefore$ Equation of circle is
$x^2+(y+b)^2=r^2$
Put $x^2=a^2-\frac{a^2 y^2}{b^2}$ in circle
$a^2-\frac{a^2 y^2}{b^2}+(y+b)^2=r^2$
$\Rightarrow\left(1-\frac{a^2}{b^2}\right) y^2+2 b y+\left(a^2+b^2-r^2\right)=0$
$D=0 \Rightarrow r^2=\frac{a^4}{a^2-b^2} \Rightarrow b=a \sqrt{1-\frac{a^2}{r^2}}$
Area of ellipse $=\pi a b$
$A=\pi a^2 \sqrt{1-\frac{a^2}{r^2}}$
$\frac{d A}{d a}=0 \Rightarrow a^2=\frac{2 r^2}{3} \Rightarrow a=\sqrt{\frac{2}{3} r}$
$\therefore \quad b=a \sqrt{1-\frac{2}{3}}=\frac{a}{\sqrt{3}}$
$\Rightarrow \quad e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{1}{3}}=\sqrt{\frac{2}{3}}$
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