Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general equation of a central hyperboloid of two sheets that is symmetric with respect to the following: (a) origin (b) \(z\) -axis (c) \(y z\) -plane

Step-by-Step Solution

Answer

\(\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = -1\)

Step 1: Identify the Symmetries

To understand the symmetries of the hyperboloid with respect to the origin, z-axis, and yz-plane, we must remember that: 1. Symmetry about the origin means if \( (x, y, z) \) is a point on the hyperboloid, then \( (-x, -y, -z) \) is also on the hyperboloid.2. Symmetry about the z-axis implies the equation is unchanged by swapping \( (x, y, z) \) with \( (-x, -y, z) \), and it often doesn't involve x and y except in their squared forms.3. Symmetry about the yz-plane implies the equation is unchanged when \( x \) changes sign, i.e., \( x \) terms must appear as \( x^2 \).

Step 2: Recall Hyperboloid Equations

A hyperboloid is given generally by the formula either as \[\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\] (one sheet)or \[\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1\] (two sheets).For this problem, we are looking for a two-sheet hyperboloid form, which indicates using the equation form with a negative right-hand side.

Step 3: Apply Symmetries to the Equation

Given the symmetries:1. Symmetry with the origin is automatically satisfied by the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1 \).2. Symmetry with the z-axis implies that the coefficients for \( x^2 \) and \( y^2 \) must be identical due to the nature of the symmetry, so \[\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = -1\].3. Symmetry with the yz-plane requires no linear \( x \) term, only \( x^2 \) appears, which is already satisfied.

Step 4: Derive the General Equation

Combining the effects of all the symmetries, the general equation of the hyperboloid becomes:\[\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = -1\].This equation translates all required symmetries into a single mathematical expression, accurately describing a central hyperboloid of two sheets.

Key Concepts

Three-Dimensional Geometry

Understanding three-dimensional geometry is crucial when dealing with objects like hyperboloids of two sheets. Three-dimensional geometry involves shapes that have length, width, and height, meaning they exist in a three-dimensional space. In this space, we can describe points using three coordinates:

• The x-coordinate, or position along the x-axis
• The y-coordinate, or position along the y-axis
• The z-coordinate, or position along the z-axis

In three-dimensional geometry, shapes such as spheres, cylinders, and hyperboloids are explored through equations that connect these three coordinates. The hyperboloid of two sheets is a specific type of quadratic surface, which can be defined by mathematical equations in terms of x, y, and z. These shapes are quintessential examples used to highlight the complexity and beauty of solid geometry. By manipulating these coordinates within the formula \[\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1\],we can characterize the structure and symmetry of such shapes in the 3D space.

Symmetry in Mathematics

Symmetry is a key concept in mathematics, especially when analyzing geometric figures like hyperboloids of two sheets. This type of symmetry helps to ensure that the object's structure looks the same from different viewpoints or transformations.
For hyperboloids:

• Origin Symmetry: Means that every point \((x, y, z)\) on the shape has a counterpart \((-x, -y, -z)\) that also lies on the shape. The hyperboloid is mirrored perfectly around the origin.
• Z-axis Symmetry: Implies the hyperboloid remains unchanged if you rotate it around the z-axis. Therefore, x and y variables appear symmetrically in the equation, often seen with \(x^2\) and \(y^2\) terms having the same coefficient.
• YZ-plane Symmetry: The equation of the hyperboloid will not alter if x changes to -x. This means only even powers of x can appear (like \(x^2\)), ensuring symmetry in relation to the yz-plane.

By ensuring these symmetries are present in an equation, we can create equations that are highly structured and predictable.

Quadratic Surfaces

Quadratic surfaces are a fascinating part of geometry and include several types of conic shapes, like ellipsoids, paraboloids, and hyperboloids. These surfaces are formed when a quadratic equation is graphed in three-dimensional space. For example, the general equation for a hyperboloid of two sheets can be written as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1\]This specific form characterizes two separate "sheets" or disconnected curved surfaces.
Quadratic surfaces like the hyperboloid of two sheets exhibit distinctive features:

• Curvature: These surfaces bend and curve, creating a complex structure within three-dimensional geometry.
• Axial Symmetry: For hyperboloids, they often exhibit symmetry about certain axes, such as the z-axis.
• Central Nature: The center location at the origin is a defining feature, making transformations and symmetry analysis easier, as seen with origin symmetry.

Hyperboloids function as real-world examples for understanding quadratic surfaces, showing the application of mathematical concepts into tangible structures.