This is a similar to a prelim question that I came across.

Find the distance between the following two ellipsoids:

$$\frac{x_1^2}{a_1}+\frac{x_2^2}{a_2}+\cdots\frac{x_n^2}{a_n}=\pi$$ and


where, $$a_1>a_2>\cdots>0$$.

My idea is to use Lagrange Multipliers, but I cannot seem to implement this idea.

The method is solving the system

$$\nabla f=\lambda\nabla g$$

$$g=c, c, \lambda\in \mathbb{R}$$, and plug back into $f$ to get the minimum and maximum.

I am quite rusty on Lagrange Multipliers, so any help would be appreciated.


You could think about it geometrically, the two surfaces are concentric and similar the minimum distance between them will be along a line going through the origin and will be smallest for the smallest intercept which will be the $x_n$ intercept for which the distance equals $\sqrt{a_n \pi}-\sqrt{a_n e }$

  • $\begingroup$ This makes complete sense, because I did this with examples and saw it. The question that I have is what would be the best way to formulate this in a rigorous mathematical manner? $\endgroup$
    – Krampus
    Dec 1 '15 at 2:01

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