<s>[INST] System of ellipses with same foci Consider a system of ellipses $E_u$ with same foci $(\pm u,0)$, here $u>0$ is a constant. In other words, we consider a system of equations
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\quad a^2-b^2=u^2.$$
Q: How to show that any ellipse in $E_u$ satisfies
$$xyy'^2+(x^2-y^2-u^2)y'-xy=0,$$
here $y'=\frac{dy}{dx}$?
 [/INST] We have $$\frac{2x}{a^2}+\frac{2yy'}{b^2}=0$$ or
$$y'=-\frac{b^2x}{a^2y}.$$
Thus, we need to prove that
$$\frac{xy\cdot b^4x^2}{a^4y^2}-\frac{(x^2-y^2-a^2+b^2)b^2x}{a^2y}-xy=0$$ or
$$x(b^2-a^2)(b^2x^2-a^2y^2-a^2b^2)=0$$ or
$$(b^2-a^2)x\left(\frac{x^2}{a^2}-\frac{y^2}{b^2}-1\right)=0.$$
Done!
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