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Chapter 11: Problem 31

The given equation represents a quadric surface whose orientation is different from that in Table \(11.7 .1 .\) Identify and sketch the surface. $$ z=\frac{x^{2}}{4}-\frac{y^{2}}{9} $$

Short Answer

Expert verified
The surface is a hyperbolic paraboloid oriented along the z-axis.

Step by step solution

01

Identify the Type of Quadric Surface

The equation \( z = \frac{x^2}{4} - \frac{y^2}{9} \) is in the form \( z = \frac{x^2}{a^2} - \frac{y^2}{b^2} \). This is the standard form of a hyperbolic paraboloid, characterized by having one term with a positive coefficient and one term with a negative coefficient.
02

Understand the Orientation

The given form \( z = \frac{x^2}{4} - \frac{y^2}{9} \) suggests the quadric surface is a hyperbolic paraboloid oriented along the z-axis, as \( z \) is isolated on one side of the equation. The squared terms of \( x \) and \( y \) determine the curvature in the \( xy \)-plane.
03

Sketch the Surface

To sketch the hyperbolic paraboloid, note that cross-sections parallel to the \( xz \)-plane (\( y = \text{constant} \)) are parabolas opening upwards (if the coefficient of \( x^2 \) is positive, downward if negative), and cross-sections parallel to the \( yz \)-plane (\( x = \text{constant} \)) are parabolas opening downwards (since the coefficient of \( y^2 \) is negative). The saddle shape is characteristic of a hyperbolic paraboloid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbolic Paraboloid
A hyperbolic paraboloid is a fascinating type of quadric surface that has a unique saddle shape. You might compare it to a Pringles chip or a riding saddle. It manifests when you have an equation like \( z = \frac{x^2}{a^2} - \frac{y^2}{b^2} \). This form is essential as it has one squared term positive and the other negative. The equation represents a surface of infinite extent that curves differently in two axial directions.
  • In the direction of the positive term (usually the \( x^2 \) term), you will have upward parabola shapes.
  • In the direction of the negative term (\( y^2 \), in this case), you'll have downward parabola shapes.
This interplay of directions forms the characteristic "saddle" curvature. Understanding this shape and equation form makes it easier to identify and work with hyperbolic paraboloids, vital for many applications in architecture and physics.
Orientation of Surfaces
The orientation of a hyperbolic paraboloid can be determined by looking at which variable is isolated on one side of the equation. In the equation \( z = \frac{x^2}{4} - \frac{y^2}{9} \), \( z \) is isolated, indicating the surface is oriented so that it primarily stretches out along the \( z \)-axis. This equation's orientation is different compared to when the\( z \)-axis isn't in an isolated position.
  • If \( z \) is by itself, we generally say the surface is "oriented along the z-axis."
  • Similarly, if \( x \) or \( y \) were isolated, it would denote orientation along those respective axes.
Understanding orientation helps when trying to visualize how the surface lies in 3D space, which fields it covers more expansively, and what type of intersections it might have with other planes or objects.
Cross Sections
Cross sections are slices of the surface made by intersecting it with planes. They provide a clearer idea of the surface's shape by breaking down its complexity. For a hyperbolic paraboloid like \( z = \frac{x^2}{4} - \frac{y^2}{9} \), here's what its cross sections might look like:
  • For sections parallel to the \( xz \)-plane, set \( y = \text{constant} \). You will get a series of upward opening parabolas, thanks to the \( x^2 \) positive term.
  • For sections parallel to the \( yz \)-plane, \( x = \text{constant} \), you will see downward-opening parabolas, due to the negative \( y^2 \) term.
Recognizing these patterns helps you predict the shape and curvature, ensuring a more robust understanding of geometric and functional applications.
Surface Sketching
Sketching a hyperbolic paraboloid might seem daunting, but breaking it down can simplify the process. Start by understanding its cross sections, as discussed above. Here's a simple guide to help:
  • Mark the center or the "saddle" point where the surface changes direction.
  • From the center, sketch the parabolas along the cross-sections you already plotted.
  • Focus on how these cross-section parabolas convey the saddle shape. Ensure the upward and downward curvatures are apparent.
Now, connect these parabolas smoothly across the surface center. Remember, the hyperbolic paraboloid expands infinitely, so its edges should taper off gently to indicate this effect. Practice makes perfect here, drawing from each cross section to get a coherent full picture.

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Most popular questions from this chapter

Sketch the surface. $$ z=\sqrt{x^{2}+y^{2}-1} $$

Discuss some of the connections between conic sections and traces of quadric surfaces.

If \(L\) is a line in 2 -space or 3 -space that passes through the points \(A\) and \(B\), then the distance from a point \(P\) to the line \(L\) is equal to the length of the component of the vector \(\overrightarrow{A P}\) that is orthogonal to the vector \(\overrightarrow{A B}\) (see the accompanying figure). Use this result to find the distance from the point \(P(1,0)\) to the line through \(A(2,-3)\) and \(B(5,1)\).

Consider the lines \(L_{1}\) and \(L_{2}\) whose symmetric equations are $$ \begin{aligned} &L_{1}: \frac{x-1}{2}=\frac{y+\frac{3}{2}}{1}=\frac{z+1}{2} \\ &L_{2}: \frac{x-4}{-1}=\frac{y-3}{-2}=\frac{z+4}{2} \end{aligned} $$ (see Exercise 52). (a) Are \(L_{1}\) and \(L_{2}\) parallel? Perpendicular? (b) Find parametric equations for \(L_{1}\) and \(L_{2}\). (c) Do \(L_{1}\) and \(L_{2}\) intersect? If so, where?

These exercises refer to the hyperbolic paraboloid \(z=y^{2}-x^{2}\) (a) Find an equation of the hyperbolic trace in the plane \(z=4\). (b) Find the vertices of the hyperbola in part (a). (c) Find the foci of the hyperbola in part (a). (d) Describe the orientation of the focal axis of the hyperbola in part (a) relative to the coordinate axes.
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