Question

The plane x + y + 2z = 10 intersects the paraboloid z = x^2 + y^2 in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.

Answer 1

gradient f = <2x, 2y, 2z>
gradient g = <1,1,2>
gradient h = <2x, 2y, -1>

2x=λ+2vx
2y=λ+2vy
2z=2λ-v
x+y+2z=10
\[x^2+y^2-z=0\]


2x-2xv=λ
2y-2yv=λ
(2z+v)/2=λ

2x-2xv=2y-2yv
2x(1-v)=2y(1-v)

either x=y or v=1

case 1: x=y
x+x+2z=16 == 2x+2z=16
x^2+x^2-z=0 == 2x^2-x=0

2x=16-2z
x=8-z

2(8-z)^2-z=0
2(z^2-16z+64)-z=0
2z^2-32z+128-z=0
2z^2-33z+128=0

not sure where to go from here or what i did wrong

case 2: v =1

2x=λ+2x λ=0
2z=2(0)-(1) 2z = -1 z = -1/2
x+y+2(-1/2) = 10
x+y=9
x=9-y

x^2+y^2-(-1/2)=0
(9-y)^2+y^2+1/2=0
y^2-1881+y^2+1/2=0
2y^2-18y+81+1/2=0

and once again im stuck here as well