1. Consider the point P with coordinates (a; b; c) on the surface of the hyperbo
ID: 3374197 • Letter: 1
Question
1. Consider the point P with coordinates (a; b; c) on the surface of the hyperbolic paraboloid described
by z = y^2-x^2
(a) Determine z(t) such that the straight line r(t) = (a + t)i + (b + t)j + z(t) k lies entirely on the
surface and passes through the point P. Simplify your result so that only the terms a, b, and c
appear. That is, terms such as a^2, or a^3, do not appear.
(b) A ruled quadric surface is a quadric surface that can be generated by the continuous motion of a
straight line. Clearly, based on your results from part (a), any point on a hyperbolic paraboloid
has at least one straight line that passes through it, but is the hyperbolic paraboloid a ruled
quadric surface? If so, what motion of a straight line will generate the hyperbolic paraboloid?
(c) Are there any other ruled quadric surfaces? If so, which surfaces are they, and how are they
generated?
Explanation / Answer
(a) z(t) = (b+t)^2 - (a+t)^2 - b^2 +2bt + t^2 - a^2 -2at - t^2 = (b^2-a^2) + 2(b-a)t
(b) Yes, it is a ruled quadric surface.
(c) There is one other: the hyperboloid of one sheet.
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