Say we have a conic with equation $f(x,y)=c$.
My teacher says that it's centre satisfies the equations :
$f_x(x,y)=f_y(x,y)=0$ (If it has a centre).
She didn't give any explanation. I thought this was because if we have centre $(x_0,y_0)$, then $f(x_0+x,y_0+y) =f(x_0-x,y_0-y)$(As it's a centre, the point diametrically opposite is on the curve). Differentiating, we get $f_x(x_0+x,y_0+y) =-f_x(x_0-x,y_0-y)$ and setting $x=y=0$, we get the result. Is this correct? I realize that I have assumed the entire family of curves $f(x,y)=c$ have the same centre,how do I avoid that? Is this valid for other curves?