Thread: Nearest and farthest point

The plane intersects the paraboloid in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.

What I did:

Let and
and

then:








But this is none of the answer. The answers are

Can someone point out my mistake?

Here you have 2 constraints


x + y + 2z = 2 call it f

z-x^2 -y^2 = 0 call it g

You want to maxmimize the distance to the origin
(maximize the square of the distance)

Call it D = x^2 + y^2 + z^2

gradD= L1 gradf + L2 gradg

This leads to the equations:

1) 2x = L1 -2xL2
2) 2y = L1 -2yL2
3) 2z= 2L1 + L2
4) x + y + 2z =2
5) z = x^2 + y^2


Now
i Eliminate L2 between eqns 1) and 2) and you obtain x = y

ii Use this in 4) to get z = 1 - x

iii Use this in 5 to get x = 1/2 or -1

Use this to finish