Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 16
\(15-18=\) Show that the equation represents a sphere, and find its center and radius. $$ x^{2}+y^{2}+z^{2}+4 x-6 y+2 z=10 $$
Short Answer
Expert verified
The sphere's center is at (-2, 3, -1) and its radius is 2鈭�6.
Step by step solution
01
Rearrange Equation into Standard Form
First, we need to express the given equation in the standard format of a sphere's equation. A sphere's equation typically has the form \[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\]We start by grouping the quadratic terms with the corresponding linear terms: \[x^2 + 4x + y^2 - 6y + z^2 + 2z = 10\]
02
Complete the Square for X-Terms
To complete the square for the terms involving \(x\), we take \(x^2 + 4x\) and add and subtract \((\frac{4}{2})^2 = 4\):\[(x^2 + 4x) = (x + 2)^2 - 4\]
03
Complete the Square for Y-Terms
For the \(y\) terms, we complete the square using the expression \(y^2 - 6y\): Add and subtract \((\frac{-6}{2})^2 = 9\):\[(y^2 - 6y) = (y - 3)^2 - 9\]
04
Complete the Square for Z-Terms
For the \(z\) terms, complete the square for \(z^2 + 2z\) by adding and subtracting \((\frac{2}{2})^2 = 1\):\[(z^2 + 2z) = (z + 1)^2 - 1\]
05
Combine Perfect Squares
Insert the completed square terms back into the equation:\[(x + 2)^2 - 4 + (y - 3)^2 - 9 + (z + 1)^2 - 1 = 10\]Combine the constants on the left side:\[(x + 2)^2 + (y - 3)^2 + (z + 1)^2 = 10 + 4 + 9 + 1\]
06
Solve for the Right Side and Identify Sphere Parameters
Simplifying the right side:\[(x + 2)^2 + (y - 3)^2 + (z + 1)^2 = 24\]This indicates the sphere has a center at \((-2, 3, -1)\) and a radius of \(\sqrt{24} = 2\sqrt{6}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a technique used to convert a quadratic expression into a perfect square trinomial. This is particularly useful in rewriting equations into standard forms, such as the equation of a sphere.
To complete the square, you follow several steps:
To complete the square, you follow several steps:
- Take the quadratic term plus the linear term, for instance, terms like \(x^2 + 4x\).
- Find the square of half of the coefficient of the linear term. Here, \(\left(\frac{4}{2}\right)^2 = 4\).
- Add and subtract this square inside the equation so that the trinomial becomes a perfect square: \((x + 2)^2 - 4\).
Standard Form of a Sphere
The equation of a sphere is typically written in its standard form to easily identify its center and radius. The standard form of a sphere's equation is:
In our case, the steps turn the original polynomial \(x^2 + y^2 + z^2 + 4x - 6y + 2z = 10\) into the standard form \((x + 2)^2 + (y - 3)^2 + (z + 1)^2 = 24\), identifying the geometrical nature of the object represented by the equation.
- \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\)
In our case, the steps turn the original polynomial \(x^2 + y^2 + z^2 + 4x - 6y + 2z = 10\) into the standard form \((x + 2)^2 + (y - 3)^2 + (z + 1)^2 = 24\), identifying the geometrical nature of the object represented by the equation.
Center of a Sphere
The center of a sphere in the standard equation is given by the coordinates \((h, k, l)\). These are extracted directly from the transformed equation \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\). For instance, in \((x+2)^2 + (y-3)^2 + (z+1)^2\), we identify the center as \((-2, 3, -1)\).
These values are deduced as being the symmetric opposite of what's within the parenthesis alongside each variable term:
These values are deduced as being the symmetric opposite of what's within the parenthesis alongside each variable term:
- For \((x + 2)\), the shift is \(-2\).
- For \((y - 3)\), the shift is \(3\).
- For \((z + 1)\), the shift is \(-1\).
Radius of a Sphere
The radius of the sphere is the distance from its center to any point on its surface. In the sphere's standard form equation \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), \(r\) represents the radius.
By looking at the equation \((x+2)^2 + (y-3)^2 + (z+1)^2 = 24\), we can deduce that \(r^2 = 24\). By taking the square root, the radius of the sphere becomes \(r = \sqrt{24}\), which simplifies to \(r = 2\sqrt{6}\).
By looking at the equation \((x+2)^2 + (y-3)^2 + (z+1)^2 = 24\), we can deduce that \(r^2 = 24\). By taking the square root, the radius of the sphere becomes \(r = \sqrt{24}\), which simplifies to \(r = 2\sqrt{6}\).
- This transformation reveals the sphere's size in three-dimensional space.
- Simplifying the expression ensures clarity in understanding the sphere's radius.
