Question

Use the method of Lagrange Multipliers to find the points on the surface ?? − ? 2 = 9 that are closest to the origin

Answer 1

Ando Р (wy, ) Pluz, 2) Р O (دردره ) so Let gren surface xy-2², and the have to find bon't on the surface of that curre kinose distance from origion is Minimum. ie Minimum (op) =) Minman (16p, ²): so Let ffa72)=x²ty²+z² subjected to xy - 2²=9. g(x,y,z 7,2) = xy-z'-9 op = Jutytz using Lagrange Multibler Method: Op??= x+y+z" of =xgg =) (x²+4 +2²) d (uy-2²-9) 2n=xy ----(i) similarly af EX oy (***y+2) => (27-29) sy 2y = > :> X - -.fi og (x++y+z") = 1 . (xy-zy) 22= *(-22)---(11) d dx dx dx siminity of a =x dg dz عل

multibly (ty and (ii): 4 xy = x xy = xy(4-x²)=0. so eitners ayzo or 4- x 20 = 7212, 4 If xy =o, then but into (ii): ny-2² = 9 0-2'=9 =)2=-9 (No Real values of a ny = oi's not bossible. so lle have x=+2; but >=12 in ci, and (ii) 2x=xy-f) =) 2x = +2y = x=ty. .-(1) -) 2 4 * == x=) * = + 3) ) cy 24 = xu 크래 21 -) > x=ty 0 ?C=0 also but x = 12 in (iii) 22=+2(-22) =) 2Z = 542 - 22+42) so entier 62=0 or 22=0 = ZO.

I but x=-4 in ay-z²9 =) - 7²0² = 9 quel Z (roreal value of of y) but h=4 and 220 in ny-z = 9 = 4²²9 =) y=+3. < = 3 anel z= 0. Р so If y=3; then se=y. so bonito are (?, % 2) = (3,3,0) similaily If y=-3. Then x=y=-3 and 2 zo. so bon't p are (1,4,2)= (-3,-3,0). • but (17,2)=( 33,0) in (op' = nety +z = 3²+ 370 18; but (1,7,2)=(-3-3.") in cop) 'n'ty' +2°= 3+2+ 18. so bon't P(r, y, z) on the surface ny-z=0 kinose distance from (0,0,0) origin is minimum is: (3-3,373 (3,3,0) (closed distance)