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Chapter 2: Problem 344

For the following exercises, the equation of a quadric surface is given. a. Use the method of completing the square to write the equation in standard form. b. Identify the surface. $$ x^{2}-y^{2}+z^{2}-12 z+2 x+37=0 $$

Short Answer

Expert verified
The surface is a hyperbolic cylinder.

Step by step solution

01

Group and Rearrange by Variable

First, rearrange and group the terms by their respective variables:\[ x^2 + 2x - y^2 + z^2 - 12z + 37 = 0 \]
02

Complete the Square for x-terms

Take the terms involving \(x\):\[ x^2+2x \]To complete the square, take half of the linear coefficient (2), square it, and add it inside the square:\[ (x+1)^2 - 1 \]
03

Complete the Square for z-terms

Take the terms involving \(z\):\[ z^2 - 12z \]Complete the square by taking half of -12, square it, then add that inside the square:\[ (z-6)^2 - 36 \]
04

Substitute Completed Squares Back

Substitute the completed squares back into the equation:\[(x+1)^2 - 1 - y^2 + (z-6)^2 - 36 + 37 = 0\]Simplify the constants:\[(x+1)^2 - y^2 + (z-6)^2 = 0\]
05

Identify the Surface

The equation \((x+1)^2 - y^2 + (z-6)^2 = 0\) describes a hyperbolic cylinder as it has a negative sign with one term and equals zero.

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Key Concepts

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

Completing the Square
Completing the square is a method widely used to manipulate quadratic equations. It involves converting a quadratic expression into a perfect square trinomial, something very helpful when dealing with quadric surfaces. Let's break it down simply.
  • To complete the square for a term like \(x^2 + bx\), first take the linear coefficient \(b\), divide it by 2, and then square it. This forms the expression \((x + \frac{b}{2})^2\).
  • For example, if we consider the expression \(x^2 + 2x\), as shown in the solution, taking half of 2 results in 1. Squaring 1 gives us 1, turning the expression into \((x+1)^2 - 1\). This makes it easy to handle the quadratics within a larger equation like the given quadric surface equation.
Following these steps correctly aids in transforming an equation into a more manageable form for further simplification, crucial for studying quadric surfaces.
Standard Form
The standard form of an equation provides a clearer view of what type of quadric surface it might represent. Achieving this often depends on completing the square for any quadratic terms in the equation. In doing so, the equation transitions into a format that can be more readily analyzed.
  • Once the quadratic parts are completed, the terms of the equation are organized, allowing us to see the structure of the surface.
  • In the solution, after completing the square for \(x^2 + 2x\) and \(z^2 - 12z\), the equation simplifies significantly: \((x+1)^2 - y^2 + (z-6)^2 = 0\). This highlights the relationships between the variable terms in a clear, structured manner.
  • This form makes it easier to identify the type of surface represented, often revealing symmetries or characteristics like centers, axes, and shapes.
Having the equation in standard form unveils the nature of the quadric surface, making it not only simpler to understand but also to graph and analyze.
Hyperbolic Cylinder
A hyperbolic cylinder is one of the many types of surfaces formed in three-dimensional space defined by a quadric equation. Identifying it involves recognizing the particular structure of its equation.
  • In the form \((x-h)^2 - (y-k)^2 + (z-l)^2 = 0\), a hyperbolic cylinder shows distinctive features. The negative sign before one of the squared terms is a dead giveaway of its hyperbolic nature.
  • For the exercise equation \((x+1)^2 - y^2 + (z-6)^2 = 0\), the presence of \(-y^2\) indicates this is indeed a hyperbolic cylinder. The equation is balanced, combining a subtraction of squares with the result reaching zero, signifying its continuous and infinite form.
  • This surface does not have a circular base, rather a hyperbolic half that defines its unique structure. It runs infinitely along one direction, unlike ellipsoids or paraboloids that enclose a space.
Understanding the characteristics of different quadric surfaces like the hyperbolic cylinder aids in comprehending their geometric configurations and how they apply in various fields, from physics to architecture.

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