Question

Use Lagrange multipliers to find the shortest distance from the point (2,0, -9) to the plane x + y + z = 1 MY NOTES ASK YOUR TEACHER 10. DETAILS SESSCALC2 11.6.049. Find parametric equations for the tangent line to the curve of Intersection of the paraboloid = x2 + y2 and the ellipsoid 3x +212 +722 - 33 at the point (-1,1,2). (Enter

Answer 1

let (rary, Z) be any point on the blame aay-12 = 1 the distemoe of (2,0-9) from the plane now x+y+z=1 is Sar (2-2) 7 f g h (2 +972 where nagrZ=1 The Lagramgian for thi's problem 20899,2). 2. W.C237 S7 (719)2 + 2(+2-1) where is an Lagrange's multiplier (x-2) (4.29% yh (Z 1912 NO- te z Ly z ta ~ (x-2)2 + 4 LZ 1922 Lz z Z 19 ta √(x-2)2 y + ( 2 1922 This gives to complecacy for find the shortest distance To find the shortest distance 30 we first find, the minimum rame of 8 = = (x-2) M ( 2+972 subje of to near Z 21 Now L(doy, z) = (2-2) 7 (26-2) 7 9 7 (219) 2+ Besty +7-4) La = 200-2) to Ly z 29 to L2 = 212+92 + and NaN+7 21 for stationary points, wehawe La = hy zlzzo od 2 cm-2) th 20 24 ta 20 21219) + 20 Thigives 2 (Aty +Z170) + 30 20 aty+z= 1 16 CS Scanned with Oalascannen +79) + 39 20' om 13 3

-16 on, i 200-2) - and 20 or, nr2 = doing my (m, 99Z) = (-3/ -19 3. doo 83 2(2-2) ta zo gives. 2 (dmt (dy)** 221%), positively defibrine. 8 52 Ryta 20 - y = 217+9) to 20 -) Z+9= o dy (od Odz)? hence At 1997 CF () - units. Now ñ 2 & Toy - ما Lauldnt byyyy -32-(02)2 2 1. . At (10, 5, ), st is the minimum and hence Shortest distance sa 10. suppose fony, Z) 27 y v z=0- The direction o ratios of the normal of the surface +20 at of of on Tom Toy) 2 ,1,2) And 301244+ Tz?. 33 4)|4,103 (dil, 2) is (239 29 -1) (-0,1,2)=(-2,23-1) gim, y, z) Similarly for 9-0 Surface, Todojamo) – (6%, 10, 147 (-6, 4,:28) og , og oil 142)|(410,2) 2 Of dn P of dyp oz da TY On Now we know direction of cutio of the dangent line to the curree of intersection of 12088=0 is given the two equations. of at (1,4, 2) on Of dat og dy toy dz zo - 20n +2dy -adz 20 2dn-2 dy raz zo sdn-ady 14dz zo odre tady +28dzzo da dy om, de manera la 28+24 csskeie direction, radios of tomgent lines at (-deb, 3are(32, 31,-2) or dz -8+6 -2 B-128

is AH 32 2-2 -2 Henoe the caucujon, of tangent line at (-1,1,2) y ! 'i the parametric equaticou of tongent line is na Bat-1 y = 21,571,2=-24+2 31 =% CS Scanned with CamScanner