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Chapter 2: Problem 326

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. \(63 x^{2}+7 y^{2}+9 z^{2}-63=0\)

Short Answer

Expert verified
The surface is an ellipsoid.

Step by step solution

01

Identify the Quadric Form

The given equation is a quadratic equation in three variables: \(63 x^{2}+7 y^{2}+9 z^{2}-63=0\). This represents a type of 3D surface known as a quadric surface.
02

Adjust Constant Term

First, move the constant term to the other side of the equation: \[63x^2 + 7y^2 + 9z^2 = 63.\] This sets up the equation for canonical form.
03

Normalize Each Term

Divide each term in the equation by 63 to simplify it: \[x^2 + \frac{7}{63}y^2 + \frac{9}{63}z^2 = 1.\] This simplifies to \[x^2 + \frac{y^2}{9} + \frac{z^2}{7} = 1.\]
04

Identify the Quadric Surface

The equation \(x^2 + \frac{y^2}{9} + \frac{z^2}{7} = 1\) is in standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\), where every denominator is positive. This is the standard form of an ellipsoid.

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Key Concepts

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

Standard Form
The standard form of a quadric surface helps simplify complex equations and identify the surface type. Quadric surfaces are represented by second-degree polynomials involving three variables, like the example equation from the exercise, containing variables \(x\), \(y\), and \(z\). To rewrite a quadric surface equation in standard form, you isolate the constant on one side and express the equation as a sum of squared terms divided by constants, equal to 1.
  • Move the constant term to create positivity on the right side: \(63x^2 + 7y^2 + 9z^2 = 63\).
  • Divide by the largest constant to normalize each term and organize it into a standardized pattern: \(x^2 + \frac{y^2}{9} + \frac{z^2}{7} = 1\).
This standardized form assists in quickly identifying the geometric nature of the surface, like determining if it's an ellipsoid, hyperboloid, or another type.
Ellipsoid
An ellipsoid is a type of quadric surface that appears in a shape similar to a stretched or squashed sphere. When an equation is converted into standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\), we identify it as an ellipsoid, provided all the denominator constants are positive. The coefficients hint at the different radii lengths along the principal axes:
  • The smaller the denominator, the larger the respective axis radius and vice versa.
  • For our exercise, the equation \(x^2 + \frac{y^2}{9} + \frac{z^2}{7} = 1\) represents an ellipsoid.
The equation shows that the ellipsoid extends further along the axis with larger denominators, shaping an irregular sphere.
Quadratic Equation
A quadratic equation in three variables is central to defining quadric surfaces. These equations form second-degree polynomials that include terms like \(x^2\), \(y^2\), and \(z^2\), and they typically associate with complex 3D shapes. The transformed equation \(x^2 + \frac{y^2}{9} + \frac{z^2}{7} = 1\) showcases how quadratic equations define quadric surfaces in relation to an ellipsoid.
  • The key aspect of these equations lies in their degree, which helps categorize the surface.
  • The normalization and rewriting process of such equations is crucial in revealing their geometric nature.
A deep understanding of these concepts aids in visualizing and interpreting the 3D shapes formed by the equations.
3D Geometry
3D geometry concerns the study of shapes and figures in three-dimensional space, an essential element in understanding quadric surfaces. When dealing with 3D geometry, you engage in analyzing surfaces like ellipsoids that extend in the x, y, and z axes.
Quadric surfaces within this domain can represent various real-world objects. For instance, ellipsoids can model celestial bodies like planets or prolate spheroids like rugby balls.
  • Understanding the geometric properties, like axis lengths from the equation's constants, highlights the object's shape.
  • The orientation and dimensions of the surface describe how it is aligned and stretched in space.
This understanding supports interpreting 3D space through equations, enhancing spatial visualization skills.

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