Calculus: Analytic Geometry and Calculus, with Vectors

404 Cones and conics

z = k in a set which is the empty set ifjkI > c, a point if Ikl = c, andan
ellipse (or circle if a = b) if Ikl < c. Very similar remarks apply to the
planes having the equations x = k and y = k. It is easiest to sketch the
intersections (or sections) for which k = 0, but full information about
other sections parallel to the coordinate planes is easily obtained. For
example, when Jkl < c, we can put z = k in (6.63), transpose the term
involving c2, and then divide by the new right side to obtain
x2 y2

a2 ( 1 -k b2 (1 -k2)

In case a b, this shows that the graph intersects the plane having the

equation z = k in an ellipse which has its center on the z axis and which

intersects the planes having the equations x = 0 and y = 0 at the points

2

(o,±b.J1_k,k), (±atJ1_,0,k).

In case a = b, the section is a circle. Unshaded and shaded graphs appear

in Figures 6.631 and 6.632. In case a = c and b is greater than a and c,

Figure 6.631 Figure 6.632

the graph is a prolate spheroid more or less like a cucumber. In case

a = b and c is less than a and b, the graph is an oblate spheroid more or

less like the earth (which is depressed at the poles and bulges at the

equator) or like a pancake. It is possible to use material from Section 2.6

and Chapter 6 to show that each plane section of an ellipsoid must be an

empty set or a single point or a circle or an ellipse.

The graph of the equation

(6.64) a2 + b2

is a hyperboloid of one sheet. It intersects the plane having the equation

z = k in an ellipse (or circle if a = b). It intersects the plane having the

equation y = k in a hyperbola when Jkl 0 b and in a pair of lines when

Jkl = b. It intersects the plane having the equation x = k in a hyperbola

when Jkl 0 a and in a pair of lines when Jkl = a. Unshaded and shaded

graphs appear in Figures 6.641 and 6.642.