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

step-by-step explanation:

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