The problem I am working on asks:
Ellipsoid $\frac{x^2}{3}+\frac{y^2}{2}+\frac{z^2}{1}=1$ intersect with plane $x+y+z=0$, what is the area of the intersection?
My idea is basically as same as this answer, and using Lagrange's multiplier to find maximum and minimum of $x^2+y^2+z^2$. So here is the function constructed:
$$L(x,y,z,\lambda,\mu)=x^2+y^2+z^2+\lambda(\frac{x^2}{3}+\frac{y^2}{2}+\frac{z^2}{1}-1)+\mu(x+y+z)$$
Then taking derivative:
$$L_x(x,y,z,\lambda,\mu)=2x+\frac{2}{3}{\lambda}x+\mu=0$$ $$L_y(x,y,z,\lambda,\mu)=2y+{\lambda}y+\mu=0$$ $$L_z(x,y,z,\lambda,\mu)=2z+2{\lambda}z+\mu=0$$ $$L_\lambda(x,y,z,\lambda,\mu)=\frac{x^2}{3}+\frac{y^2}{2}+\frac{z^2}{1}-1=0$$ $$L_\mu(x,y,z,\lambda,\mu)=x+y+z=0$$
I failed to find the solution of this on my self, so I had to find answer in the solution manual. It says "Observe this five equation, we can learn that $\lambda+x^2+y^2+z^2=0$". But I am quite confused where this conclusion come from, I tried to do something like eliminate $\lambda$ from these equations but get nothing. Also, I do not understand why in this particular case, the Lagrange's multiplier $\lambda$ is equal to the opposite of the function being optimized (i.e. $x^2+y^2+z^2$), will this also works on any ellipsoid intersect with any plane? Could anyone expand this in detail?
Things below, I believe is not related to my problem, however, I think I'd better describe the full solution in case anyone find it useful. Eliminate $\mu$ from first three equations, combine with the last equation, we get $\left[ \begin{array}{ccc} 6+2\lambda&-6-3\lambda&0\\ 0&2+\lambda&-2-2\lambda\\ 1&1&1 \end{array} \right]\left[\begin{array}{c} x\\y\\z \end{array}\right]=\left[\begin{array}{c} 0\\0\\0 \end{array}\right]$, because none of $x$, $y$, $z$ equal zero (otherwise the fourth equation is not satisfied) the determinate of the matrix must be $0$, using Vieta theorem, $\lambda_1\lambda_2=3$. Because of the equation I am asking, two of the half-axis equals $\sqrt{|\lambda_1|}$ and $\sqrt{|\lambda_2|}$, the area is found.
Hint:
a simpler way is to note that the major axis of the ellipse is on the line of maximum slope of the plane, and the minor axis is un the orthogonal line passing through the center (that is obviously the origin)