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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

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

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.

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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 $\sqrt{a_n \pi}-\sqrt{a_n e }$

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  • $\begingroup$ 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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