   Chapter 12.6, Problem 51E

Chapter
Section
Textbook Problem

# Show that if the point (a, b, c) lies on the hyperbolic paraboloid z = y2 − x2, then the lines with parametric equations x = a + t, y = b + t, z = c + 2(b − a)t and x = a + t, y = b − t, z = c − 2(b + a)t 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 generating lines. The only other quadric surfaces that are ruled surfaces are cylinders, cones, and hyperboloids of one sheet.)

To determine

To show: The point (a,b,c) , parametric equations x=a+t,y=b+t,z=c+2(ba)t and x=a+t,y=bt,z=c2(b+a)t are lies on the hyperbolic paraboloid that is z=y2x2 .

Explanation

Given data:

Parametric equations are x=a+t,y=b+t,z=c+2(ba)t and x=a+t,y=bt,z=c2(b+a)t .

Solution:

Consider the given equation of hyperbolic paraboloid.

z=y2x2 (1)

For (a,b,c) ,

Substitute c for z, b for y, and a for x,

c=b2a2 (2)

For x=a+t,y=b+t,z=c+2(ba)t ,

Substitute a+t for x, b+t for y, and c+2(ba)t for z in equation (1),

c+2(ba)t=(b+t)2(a+t)2c+2(ba)t=b2+2bt+t2(a2+2at+t2)c+2(ba)t=b2+2bt+t2a22att2c=b2+2bta22at2(ba)t

c=b2+2(ba)ta22(ba)t

c=b2a2 (3)

For x=a+t,y=bt,z=c2(b+a)t ,

Substitute a

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