Show that if the point lies on the hyperbolic paraboloid , then the lines with

Chapter 12, Problem 49

(choose chapter or problem)

Show that if the point lies on the hyperbolic paraboloid , then the lines with parametric equations , , and , , both lie entirely on this paraboloid. (This shows that the hyperbolic paraboloid is what is called a ruled surface; that is, it can be generated by the motion of a straight line. In fact, this exercise shows that through each point on the hyperbolic paraboloid there are two _a, b, c_ z _ y2 _ x2 x _ a _ t y _ b _ t z _ c _ 2_b _ a_t x_ a _ t y _ b _ t z _ c _ 2_b _ a_t generating lines. The only other quadric surfaces that are ruled surfaces are cylinders, cones, and hyperbo loids of one sheet.)