example needing two Lagrange multipliers

Find the semi-axes of the ellipsePlanetmathPlanetmath of intersectionMathworldPlanetmath, formed when the plane  z=x+y  intersects the ellipsoidMathworldPlanetmathPlanetmath


Let  (x,y,z)  be any point of the ellipsoid.  The square (http://planetmath.org/SquareOfNumber) x2+y2+z2 of the distance of this point from the midpointMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/Midpoint3)  (0, 0, 0)  has under the constraints

{g:=x24+y25+z225-1= 0,h:=x+y-z= 0 (1)

the minimum and maximum values at the end pointsPlanetmathPlanetmath of the semi-axes of the ellipse.  Since we have two constraints, we must take equally many Lagrange multipliers, λ and μ.  A necessary condition of the extremumsMathworldPlanetmath of


is that in to (1), also the equations

{fx+λgx+μhx= 2x+12xλ+μ= 0,fy+λgy+μhy= 2y+25yλ+μ= 0,fz+λgz+μhz= 2z+225zλ-μ= 0, (2)

are satisfied.  I.e., we have five equations (1), (2) and five unknowns λ, μ, x, y, z.

The equations (2) give


which expressions may be put into the equation  h=0, and so on.  One obtains the values


with which the extremum points  (x,y,z) can be evaluated.  The corresponding values of fare 10 and 7517, whence the major semi-axis is 10 and the minor semi-axis 525517.

Title example needing two Lagrange multipliers
Canonical name ExampleNeedingTwoLagrangeMultipliers
Date of creation 2013-03-22 18:48:18
Last modified on 2013-03-22 18:48:18
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 51N20
Classification msc 26B10
Synonym using Lagrange multipliers to find semi-axes
Related topic ExampleOfLagrangeMultipliers
Related topic ExampleOfUsingLagrangeMultipliers