Find x, y, z maximum and minimum points of rotated ellipsoid

Question

I have a question very similar to a previously asked question: [Link]Find $x$, $y$, $z$ maximum and minimum points of ellipsoid. However, I have an ellipsoid in the form: $ Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz - 1 = 0 $. I can't figure out how to use the same method with the $ x $, $ y $, and $ z $ terms. Any help would be greatly appreciated!

Answer 1

In the previous problem, the ellipsoid has the center at $(0,0,0)$. What would happen if you move the center to $(x_c,y_c,z_c)$? The previous equation would become $$a(x-x_c)^2+b(y-y_c)^2+c(z-z_c)^2+2h(x-x_c)(y-y_c)+2g(x-x_c)(z-z_c)+2f(y-y_c)(z-z_c)-1=0$$

Expanding the products and grouping the terms will allow you to write $a,b,c,h,g,f,x_c,y_c,z_c$ in terms of $A,B,C,D,E,F,G,H,I$. Then use the result from the previous problem, and add the corresponding $x_c$, $y_c$, or $z_c$.