The plane x + y + 2z = 12 intersects the paraboloid z = x2 + y2 in an ellipse. find the points on the ellipse that are nearest to and farthest from the origin. nearest point (x, y, z) = farthest point (x, y, z) =

The distance between a point and the origin is . But since , both and have the same critical points, so we can consider instead the squared distance, .

We're looking for the extrema of subject to and . The Lagrangian is

with critical points where the partial derivatives vanish:

From the first two equations, it follows that .

Then in the last two equations,

If , then ; if , then .

So there are two critical points, and .

Let . We have a minimum distance of and maximum distance of .

-2 is the right