EXAMPLE 5 Sketch the surface z = 5y? – 4x2. 5y2 – 4k2 which open upward. SOLUTION The traces in the vertical planes x = k are the parabolas z = The traces in y = k are the parabolas z = which open downward. The horizontal traces are = k, a family of hyperbolas. We draw the family of traces in Figure 2, and we show how the traces appear when placed in their correct planes in Figure 3. In Figure 1 we fit together the terms to form the surface z = 5y? – 4x2, a hyperbolic paraboloid. Notice that the shape of the surface near the origin resembles that of a saddle. This surface will be investigated further in a later section when we discuss saddle points.

EXAMPLE 5 Sketch the surface z = 5y? – 4x2. 5y2 – 4k2 which open upward. SOLUTION The traces in the vertical planes x = k are the parabolas z = The traces in y = k are the parabolas z = which open downward. The horizontal traces are = k, a family of hyperbolas. We draw the family of traces in Figure 2, and we show how the traces appear when placed in their correct planes in Figure 3. In Figure 1 we fit together the terms to form the surface z = 5y? – 4x2, a hyperbolic paraboloid. Notice that the shape of the surface near the origin resembles that of a saddle. This surface will be investigated further in a later section when we discuss saddle points.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.3: Hyperbolas

Problem 78E

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Similar questions

An ellipse is the set of all points in the plane for which the ____ of the distances from two fixed points F1 and F2 is constant. The points F1 and F2 are called the ____ of the ellipse.
A hyperbola is the set of all points in the plane for which the ___ of the distances from two fixed points F1 and F2 is constant. The points F1 and F2 are called the ___ of the hyperbola.
Label the focus, directrix, and vertex on the graph given for the parabolas in Exercise 2 and 3. x2=12y y2=12x
By definition, a hyperbola is the locus of points whose positive difference of distances from two fixed points F1 and F2 called foci is constant. In the grid provided, find points whose difference of distances from points F15, 0 and F2-5, 0 is 6. That is, locate some points for which PF1-PF2=6 or PF2-PF1=6; point P3, 0 is one such point. Then sketch the hyperbola.