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Putnam 1938 B6

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August 20, 2021
Putnam

Problem Statement

What is the shortest distance between the plane Ax+By+Cz+1=0Ax + By + Cz + 1 = 0 and the ellipsoid x2a2+y2b2+z2c2=1.\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1. You may find it convenient to use the notation h=(A2+B2+C2)−12,m=(a2A2+b2B2+c2C2)12.h = (A^2 + B^2 + C^2)^{\frac{-1}{2}}, m = (a^2A^2 + b^2B^2 + c^2C^2)^{\frac{1}{2}}. What is the algebraic condition for the plane not to intersect the ellipsoid?
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