# example needing two Lagrange multipliers

Find the semi-axes of the ellipse of intersection, formed when the plane  $z=x\!+\!y$  intersects the ellipsoid

 $\frac{x^{2}}{4}+\frac{y^{2}}{5}+\frac{z^{2}}{25}=1.$

Let  $(x,\,y,\,z)$  be any point of the ellipsoid.  The square (http://planetmath.org/SquareOfNumber) $x^{2}\!+\!y^{2}\!+\!z^{2}$ of the distance of this point from the midpoint (http://planetmath.org/Midpoint3)  $(0,\,0,\,0)$  has under the constraints

 $\displaystyle\begin{cases}g\;:=\;\frac{x^{2}}{4}+\frac{y^{2}}{5}+\frac{z^{2}}{% 25}-1\;=\;0,\\ h\;:=\;x+y-z\;=\;0\end{cases}$ (1)

the minimum and maximum values at the end points of the semi-axes of the ellipse.  Since we have two constraints, we must take equally many Lagrange multipliers, $\lambda$ and $\mu$.  A necessary condition of the extremums of

 $f\;:=\,x^{2}\!+\!y^{2}\!+\!z^{2}$

is that in to (1), also the equations

 $\displaystyle\begin{cases}\frac{\partial f}{\partial x}+\lambda\frac{\partial g% }{\partial x}+\mu\frac{\partial h}{\partial x}\;=\;2x+\frac{1}{2}x\lambda+\mu% \;=\;0,\\ \frac{\partial f}{\partial y}+\lambda\frac{\partial g}{\partial y}+\mu\frac{% \partial h}{\partial y}\;=\;2y+\frac{2}{5}y\lambda+\mu\;=\;0,\\ \frac{\partial f}{\partial z}+\lambda\frac{\partial g}{\partial z}+\mu\frac{% \partial h}{\partial z}\;=\;2z+\frac{2}{25}z\lambda-\mu\;=\;0,\end{cases}$ (2)

are satisfied.  I.e., we have five equations (1), (2) and five unknowns $\lambda$, $\mu$, $x$, $y$, $z$.

The equations (2) give

 $x\;=\;-\frac{2\mu}{\lambda\!+\!4},\quad y\;=\;-\frac{5\mu}{2\lambda\!+\!10},% \quad z\;=\;\frac{25\mu}{2\lambda\!+\!50},$

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

 $\lambda_{1}=-10,\quad\lambda_{2}=-\frac{75}{17},\quad\mu_{1}=\pm 6\sqrt{\frac{% 5}{19}},\quad\mu_{2}=\pm\frac{140}{17\sqrt{646}}$

with which the extremum points  $(x,\,y,\,z)$ can be evaluated.  The corresponding values of $f$are 10 and $\frac{75}{17}$, whence the major semi-axis is $\sqrt{10}$ and the minor semi-axis $\frac{5\sqrt{255}}{17}$.

Title example needing two Lagrange multipliers ExampleNeedingTwoLagrangeMultipliers 2013-03-22 18:48:18 2013-03-22 18:48:18 pahio (2872) pahio (2872) 7 pahio (2872) Example msc 51N20 msc 26B10 using Lagrange multipliers to find semi-axes ExampleOfLagrangeMultipliers ExampleOfUsingLagrangeMultipliers