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)  $(0,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}0})$  has under the constraints

$\{\begin{array}{cc}g:=\frac{x^2}{4}+\frac{y^2}{5}+\frac{z^2}{25}-1=\mathrm{\hspace{0.33em}0},\hfill & \\ h:=x+y-z=\mathrm{\hspace{0.33em}0}\hfill & \end{array}$(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

$\{\begin{array}{cc}\frac{\partial f}{\partial x}+\lambda \frac{\partial g}{\partial x}+\mu \frac{\partial h}{\partial x}=\mathrm{\hspace{0.33em}2}x+\frac{1}{2}x\lambda +\mu =\mathrm{\hspace{0.33em}0},\hfill & \\ \frac{\partial f}{\partial y}+\lambda \frac{\partial g}{\partial y}+\mu \frac{\partial h}{\partial y}=\mathrm{\hspace{0.33em}2}y+\frac{2}{5}y\lambda +\mu =\mathrm{\hspace{0.33em}0},\hfill & \\ \frac{\partial f}{\partial z}+\lambda \frac{\partial g}{\partial z}+\mu \frac{\partial h}{\partial z}=\mathrm{\hspace{0.33em}2}z+\frac{2}{25}z\lambda -\mu =\mathrm{\hspace{0.33em}0},\hfill & \end{array}$(2)

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

The equations (2) give

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

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

$${\lambda }_1=-10,{\lambda }_2=-\frac{75}{17},{\mu }_1=\pm 6\sqrt{\frac{5}{19}},{\mu }_2=\pm \frac{140}{17\sqrt{646}}$$

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 $\frac{5\sqrt{255}}{17}$.