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.