Finding the center of an ellipse I read that one can calculte the center of an ellipse by calculating the partial diverative.

Let $g(x,y)=0$ be an ellipse then the center can be found by solving $\nabla g=0$

I would like to see a proof of the Lemma above (Not about general quadratic forms)
Would appreciate your help.

Consider a conic with equation
$$
g(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f=0
$$
Then $\nabla g=(2ax+2by+2d,2bx+2cy+2e)$ and this equals zero when
$$
\begin{cases}
ax+by+d=0\\
bx+cy+e=0
\end{cases}
$$
This system has a unique solution if and only if $ac-b^2\ne0$ (which is the case if the conic is an ellipse or a hyperbola). If this holds and $(h,k)$ is the solution, perform the translation
$$
\begin{cases}
x=X+h\\
y=Y+k
\end{cases}
$$
to find that the equation becomes
$$
aX^2+2bXY+cY^2+p=0
$$
which is the equation of a conic with its center at the origin. Thus $(h,k)$ is indeed the center of the conic. If $p=0$, the conic is degenerate.
