C
the
Vg
ants we
also lie in bo
Example 5 Finding Extremes of Distance on an Elipse
The plane x F y
11. 65). Find the
Solution we find the extreme values of
(the square of the distance from (r, y, 2) to the origin) subject to the c
The gradient equation in Equations (2)then gives
Vf=Avg Hvg
2i+
D+ u( +i+k
2+2y
2x=2/x + u, 2y=2AY+ H
The scalar equations in (5) yield
x=2Ax+2z→(1-A)
2y+22→(1-Ay
Equations (6)are satisfied simultaneously if either A=1 and z-0 or A+ 1
z/(
0. then solving B
ons (3)and (4)simultaneously to find the cor
responding points on the ellipse gives the two points (1,0,0)and (0, 1, 0)
This makes sense when you look at Figure 11.65
If x=y, then Equations (3)and (4)give
The corresponding points on the ellipse are
Pu
va)B-(-学1+v
Here we need to be careful, however, Although Pi and P both give local m
of f on the ellipse, P2 is farther from the origin than P
ere
the
