The hyperbolic paraboloid z = y2− x2 is an example of a ruled surface: Through each point (a, b, b2− a2) on the surface there is a straight line that lies entirely in the surface. In other words, the surface can be swept out by a line moving through space; it can be “drawn” with a “ruler.”
(a) Show that for any choice of (a, b), the plane y − x =b − a parallel to the z- axis intersects the surface z = y2− x2 in a straight line. Can you find another such plane?
(b) Use a computer algebra system to draw the portion of the surface z = y2 − x2 corresponding to −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1. Then superimpose on this graph a portion of the plane from (a) corresponding to (a, b)= (1/2, 1/3). Make a visual verification.
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