Unformatted text preview: lanes
x k and y k are
hyperbolas if k 0 but are
pairs of lines if k 0. x2
a2 Hyperboloid of One Sheet
z y2
b2 z2
c2 1 Horizontal traces are ellipses. Horizontal traces are ellipses. Vertical traces are parabolas. Vertical traces are hyperbolas. The variable raised to the
ﬁrst power indicates the axis
of the paraboloid.
x x y The axis of symmetry
corresponds to the variable
whose coefﬁcient is negative. y z
c Hyperbolic Paraboloid
z x2
a2 y2
b2 Hyperboloid of Two Sheets
z x2
a2 y2
b2 z2
c2 1 Horizontal traces are
hyperbolas.
Vertical traces are parabolas.
y x Horizontal traces in z
ellipses if k c or k Vertical traces are hyperbolas. The case where c
illustrated. 0 is x y k are
c. The two minus signs indicate
two sheets. 5E13(pp 868877) 1/18/06 11:36 AM Page 873 S ECTION 13.6 CYLINDERS AND QUADRIC SURFACES In Module 13.6B you can see how
changing a, b, and c in Table 1 affects
the shape of the quadric surface. EXAMPLE 7 Identify and sketch the surface 4 x 2
SOLUTION Dividing by y2 2z2 ❙❙❙❙ 873 0. 4 4, we ﬁrst put the equation in standard form:
y2
4 x2 z2
2 1 Comparing this equation with Table 1, we see that it represents a hyperboloid of two
sheets, the only difference being that in this case the axis of the hyperboloid is the
yaxis. The traces in the xy and y zplanes are the hyperbolas
x2
z
(0, _2, 0)
0
y y2
4 z 1 x2 z2 2 FIGURE 10 1 2 k
4 4≈¥[email protected]+4=0 y2
4 and z2
2 1 x The surface has no trace in the x zplane, but traces in the vertical planes y
k
2 are the ellipses
z2
k2
x2
1
yk
2
4
which can be written as (0, 2, 0) x 0 1 k
4 2 y 0
k for k 1 These traces are used to make the sketch in Figure 10. z EXAMPLE 8 Classify the quadric surface x 2 2z2 6x y 0. 10 SOLUTION By completing the square we rewrite the equation as
0 y 1 x 3 2 2z2 y Comparing this equation with Table 1, we see that it represents an elliptic paraboloid.
Here, however, the axis of the paraboloid is parallel to the yaxis, and it has been shifted
so that its vertex is the point 3, 1, 0 . The traces in the plane y k k 1 are the
ellipses
x 3 2...
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Full Document
 Fall '09
 hamrick
 Linear Algebra, Vectors, Vector Space, Dot Product, (pp 828837)

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