Find the length of the shortest curve between (1, 0, 0) and (1, 1, 1) on the surface x2 + y2 = z2 + 1. Be sure to explain why your answer is correct.

Question

Find the length of the shortest curve between (1, 0, 0) and (1, 1, 1) on the surface x2 + y2 = z2 + 1. Be sure to explain why your answer is correct.

Answer 1

The paraboloid x2 + y2 = z2 + 1 is a doubly ruled surface, which means that it can be swept out by a moving

line in space in two different ways. We can prove that. But lets verify the fact that the line defined by the two

given points lies entirely on the surface of the paraboloid.

The vector valued function that describes the straight line connecting the points (1, 0, 0) and (1, 1, 1) is

r ( t) = < 1, t, t > , t ∈ ℜ

and you can tell by inspection that satisfies the equation of the paraboloid

The length of the line segment connecting the points is √2