The plane x + y + 2z = 30 intersects the paraboloid z = x2 + y2 in an ellipse. Find the points on the ellipse that are nearest to and farthest from the origin. g

Complete the square to get the equation of the ellipse,

Parameterize this intersection by the vector-valued function,

where .

The distance between the origin and any point on the curve is , which is

Recall that a function and its square root share the same critical points, so we can make things easier by looking for the critical points of . Let

Differentiate and find the critical points:

The distance from the origin to the critical points of are then

Then the point closest to the origin is

and the point furthest from the origin is

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50