Calculus: Analytic Geometry and Calculus, with Vectors

406 Cones and conics

The graph of the equation

x2 V2

16.66) a2 b2 =z

is an elliptic paraboloid. It intersects the plane having the equation

z = k in the empty setwhen k < 0, in a point when k = 0, and in an

ellipse (or circle) when k > 0. It intersects the planes having the

equations x = k and y = k in parabolas. Unshaded and shaded graphs

appear in Figures 6.661 and 6.662.

Figure 6.661

The graph of the equation

(6.67)

y2 x2

b2

a2 z

Figure 6.662

is a hyperbolic paraboloid. It intersects the plane having the equation

z = k in a hyperbola when k < 0, in a pair of lines when k = 0, and in a

hyperbola when k > 0. It intersects the planes having the equations

x = k and y = k in parabolas. Unshaded and shaded graphs appear in

Figures 6.671 and 6.672. Hyperbolic paraboloids are the simplest

Figure 6.671 Figure 6.672