Exercise 1596Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Distance from origin to an ellipsoid

#Lagrange#ellipsoid#distance

Statement

Find the maximum distance from the origin to the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ with $0<a<b<c$ .

Hint

Maximize $x^2+y^2+z^2$ under the constraint; use Lagrange.

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Lagrange. $2x=\lambda\cdot 2x/a^2$ , etc. Nontrivial solutions require $\lambda=a^2$ or $b^2$ or $c^2$ .

Points. If $\lambda=a^2$ , then $y=z=0$ , $x=\pm a$ : distance $a$ . Similarly for $b,c$ .

Conclusion. Maximum distance $c$ (longest semi-axis), minimum $a$ .