Suppose $P$ is any point on the conic and $S_1, S_2$ are the foci of the conic, then the maximum value of $\left(P S_1+P S_2\right)$ is
We have,
$e x^2+\pi y^2-2 e^2 x-2 \pi^2 y+e^3+\pi^3=\pi e$
$\Rightarrow e x^2-2 e^2 x+\pi y^2-2 \pi^2 y=\pi e-e^3-\pi^3$
$\Rightarrow e\left(x^2-2 e x+e^2\right)+\pi\left(y^2-2 \pi y+\pi^2\right)$
$=\pi e-e^3-\pi^3+e^3+\pi^3$
$\Rightarrow e(x-e)^2+\pi(y-\pi)^2=\pi e$
$\Rightarrow \frac{(x-e)^2}{(\sqrt{\pi})^2}+\frac{(y-\pi)^2}{(\sqrt{e})^2}=1$
$\therefore \quad \sqrt{\pi}>\sqrt{e}$
$\therefore \quad P S_1+P S_2=2 a$
$\therefore \quad P S_1+P S_2=2 \sqrt{\pi}$
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