Chapter 13

Vertical traces are parabolas the variable raised to

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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. 5E-13(pp 868-877) 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 y-axis. The traces in the xy- and y z-planes 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 z-plane, 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 y-axis, 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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