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Chapter 11: Problem 18

Identify and sketch the quadric surface. \(v^{2} \quad q^{2}\) $$ x^{2}+y^{2}-z^{2}=9 $$

Short Answer

Expert verified
This is a hyperboloid of one sheet symmetric around the z-axis.

Step by step solution

01

Identify the Equation Type

The given equation is of the form \( x^2 + y^2 - z^2 = 9 \). This equation is a type of quadric surface known as a hyperboloid. Specifically, it is a hyperboloid of one sheet because the squares of two variables (\(x^2\) and \(y^2\)) have the same sign, and \(z\) has the opposite sign.
02

Standard Form Verification

Compare the equation to the standard form of a hyperboloid of one sheet, which is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \). Here, we can rewrite the given equation as \( \frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{9} = 1 \), indicating it matches the standard form with \( a^2 = b^2 = c^2 = 9 \).
03

Axis and Symmetry Determination

Since \(a^2 = b^2\), the hyperboloid is symmetric around the \(z\)-axis. The symmetry of variables \(x\) and \(y\) with \(a = b\) indicates it is rotationally symmetric around \(z\).
04

Sketch the Hyperboloid

To sketch, recognize the structure as resembling a distorted cylinder with a "waist." Draw the sections: circles in the \(xy\)-plane at constant values for \(z \) within the limits as \(-3 < z < 3\), and hyperbolas in the \(xz\)- and \(yz\)-planes. With each increasing \(z\), the radius of these circles increases.

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Key Concepts

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

Hyperboloid of One Sheet
A hyperboloid of one sheet is a fascinating type of quadric surface. It stands out due to its unique shape, which resembles a saddle or an hourglass. Think of it as a stretched-out cylinder that is pinched in the middle. The defining feature of this surface is how its equation is set up:
  • It has two squared terms with the same sign (like \(x^2\) and \(y^2\)).
  • A third squared term has the opposite sign (such as \(-z^2\)).
This combination of terms is exactly what we see in the equation \(x^2 + y^2 - z^2 = 9\), making it clear that we have a hyperboloid of one sheet on our hands. This type of surface is continuous and extends infinitely in all directions, giving it a smooth, round appearance.
Standard Equation Form
The standard equation form of a hyperboloid of one sheet is crucial for identifying and analyzing these surfaces. It is represented as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\]Here鈥檚 what each part means:
  • \(a^2\), \(b^2\), and \(c^2\) are constants that determine the shape and size of the hyperboloid.
  • When \(a = b\), the cross-sections of the hyperboloid are circles, indicating symmetry.
  • Each variable divided by these constants indicates the relative stretching along that axis.
In our specific example of \(x^2 + y^2 - z^2 = 9\), we rewrite it to fit this standard form by dividing everything by 9, giving us \(\frac{x^2}{9} + \frac{y^2}{9} - \frac{z^2}{9} = 1\). This confirms how each squared term contributes equally to the surface鈥檚 geometry.
Symmetric Axis
A hyperboloid's symmetric axis is vital for understanding its orientation in space. Since the equation \(x^2 + y^2 - z^2 = 9\) has equal coefficients for \(x^2\) and \(y^2\), we know the symmetry is around the third variable's axis, which in this case is the \(z\)-axis.This means:
  • The hyperboloid spins around the \(z\)-axis, giving it a rotational symmetry.
  • Cross-sections perpendicular to the \(z\)-axis form circles, indicating uniformity in those planes.
Understanding this symmetry helps when visualizing and sketching, as it suggests we should consider circles in the \(xy\)-plane that expand outward moving along the \(z\)-axis.
Graphing Quadric Surfaces
Graphing a quadric surface like a hyperboloid of one sheet involves understanding its structure and how to represent it visually.To sketch the equation \(x^2 + y^2 - z^2 = 9\):
  • Start by considering cross-sections in various planes: the plane \(z = 0\) gives a circle \(x^2 + y^2 = 9\).
  • In the \(xz\)- and \(yz\)-planes, you get hyperbolas, which show how the curve pinches inward.
  • Visualize horizontal circular cross-sections expanding as \(z\) moves farther from zero, forming the hyperboloid's distinctive shape.
This method highlights the quadric surface's complexity by considering both its symmetrical and curved features, fully capturing the structure of the hyperboloid of one sheet.

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Most popular questions from this chapter

Convert from spherical to rectangular coordinates. (a) \((1,2 \pi / 3,3 \pi / 4)\) (b) \((3,7 \pi / 4,5 \pi / 6)\) (c) \((8, \pi / 6, \pi / 4)\) (d) \((4, \pi / 2, \pi / 3)\)

For each \(x\) in \((-\infty,+\infty)\), let \(\mathbf{u}(x)\) be the vector from the origin to the point \(P(x, y)\) on the curve \(y=x^{2}+1\), and \(\mathbf{v}(x)\) the vector from the origin to the point \(Q(x, y)\) on the line \(y=-x-1\) (a) Use a CAS to find, to the nearest degree, the minimum angle between \(\mathbf{u}(x)\) and \(\mathbf{v}(x)\) for \(x\) in \((-\infty,+\infty)\). (b) Determine whether there are any real values of \(x\) for which \(\mathbf{u}(x)\) and \(\mathbf{v}(x)\) are orthogonal.

Determine whether the statement is true or false. Explain your answer. If \(\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}\) and \(\mathbf{a} \neq \mathbf{0}\), then \(\mathbf{b}=\mathbf{c}\)

Give a sequence of steps for determining the type of quadric surface that is associated with a quadratic equation in \(x, y\), and \(z\).

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