for-the-extremum-values-of-x-2-y-2-z-2-subject-to-the-constraints-ax-2-by-2-cz-2-1-and-lx-my-nz-0-show-that-the-stationary-points-setisfy-the-realation-l-2-1-a-1-m-2-1-b-1-n-2-

Question Number 133010 by rs4089 last updated on 18/Feb/21

f o r t h e e x t r e m u m v a l u e s o f x^{2} + y^{2} + z^{2} s u b j e c t t o t h e c o n s t r a i n t s

{a x}^{2} + {b y}^{2} + {c z}^{2} = 1 a n d l x + m y + n z = 0 .

s h o w t h a t t h e s t a t i o n a r y p o i n t s s e t i s f y t h e r e a l a t i o n

\frac{l^{2}}{1 + a λ_{1}} + \frac{m^{2}}{1 + b λ_{1}} + \frac{n^{2}}{1 + c λ_{1}} = 0