Find extreme values of ellipse I have the curve equation
$$
Ax^2 + By^2 + Cxy = 1
$$
which represents ellipse with center in (0, 0) and rotated some angle.

How can I find max X and Y values (not semi-axes) on this ellipse? Points are marked on the picture.
There x-es and y-es can not be moved left or right side, so I cannot find dx/dy and dy/dx.

Another way, use polar co-ordinates $x=r \cos t, y= r \sin t$ to get
$r^2(t)=\frac{1}{A \cos^2 t+ B\sin ^2 t+C \sin t \cos t}$
Let $u(t)=1/r^2(t)$ $\implies u(t)-\frac{A+B}{2}= \frac{A-B}{2} \cos 2t + \frac{c}{2}\sin 2t= \sqrt{\frac{(A-B)^2}{4}+\frac{c^2}{4}}~ \sin(2t+k)$
$\implies \frac{A+B}{2}-\sqrt{\frac{(A-B)^2}{4}+\frac{c^2}{4}}\le u(t) \le \frac{A+B}{2}+\sqrt{\frac{(A-B)^2}{4}+\frac{c^2}{4}}.$
From here one can get $r_{min}$ and $r_{max}$.
