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Magazine Chapter 9: Problem 15
Center and Radius of a Sphere Show that the equation represents a sphere, and find its center and radius. $$x^{2}+y^{2}+z^{2}-10 x+2 y+8 z=9$$
Short Answer
Expert verified
The sphere's center is (5, -1, -4) and its radius is \(\sqrt{51}\).
Step by step solution
01
Identify the form of a sphere's equation
A sphere's equation generally takes the form \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), where \((h, k, l)\) is the center and \(r\) is the radius.
02
Rearrange the equation
Given the equation is \(x^{2}+y^{2}+z^{2}-10x+2y+8z=9\). We will rearrange it to help with completing the square: \(x^2 - 10x + y^2 + 2y + z^2 + 8z = 9\).
03
Complete the square for x
Identify the \(x\) variable terms: \(x^2 - 10x\). To complete the square, add and subtract \((\frac{-10}{2})^2 = 25\). This transforms the expression to \((x-5)^2 - 25\).
04
Complete the square for y
For \(y\), identify the terms: \(y^2 + 2y\). Add and subtract \((\frac{2}{2})^2 = 1\) to get the completed square form: \((y+1)^2 - 1\).
05
Complete the square for z
For \(z\), take the terms \(z^2 + 8z\). Add and subtract \((\frac{8}{2})^2 = 16\), resulting in \((z+4)^2 - 16\).
06
Form the completed square equation
Combine these complete squares: \((x-5)^2 - 25 + (y+1)^2 - 1 + (z+4)^2 - 16 = 9\).
07
Simplify the equation
Bring constant terms to one side: \((x-5)^2 + (y+1)^2 + (z+4)^2 = 9 + 25 + 1 + 16\) simplifies to \((x-5)^2 + (y+1)^2 + (z+4)^2 = 51\).
08
Identify the center and radius
From the equation \((x-5)^2 + (y+1)^2 + (z+4)^2 = 51\), we see the center is \((5, -1, -4)\) and the radius is \(\sqrt{51}\).
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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 technique used to transform a quadratic expression into a perfect square trinomial. This is particularly useful in geometry and algebra, especially when working with the equation of a sphere. Here is how you can do it step by step:
- Identify which quadratic term to complete: For each of the variables (x, y, z) involved in the equation, focus on their specific quadratic terms. For example, in the expression \(x^2 - 10x\), you will work with the \(x\) terms.
- Find the number to add: Take the coefficient of the linear term (beside the variable), divide it by 2, and then square the result. For \(x^2 - 10x\), divide \(-10\) by 2 which is \(-5\), and square it to get 25.
- Add and subtract this number: To maintain the equation's balance, add and subtract the squared number. So, you create \((x-5)^2 - 25\) from \(x^2 - 10x\).
Center of a Sphere
The center of a sphere in 3D space is the point equidistant from all points on the surface of the sphere. Using the standard form of a sphere's equation, o \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\),we can identify the center:
- Look within each completed square: After completing each square for x, y, and z terms, the center coordinates \((h, k, l)\) arise. These are the constants subtracted inside each squared term.
- Extract the center values: The expression \((x-5)^2 + (y+1)^2 + (z+4)^2 = 51\) indicates the sphere's center at \((5, -1, -4)\). Here, 5 is the x-coordinate, -1 the y-coordinate, and -4 the z-coordinate.
Radius of a Sphere
The radius of a sphere is the fixed distance from the sphere's center to any point on its surface. It is a vital geometric attribute that defines the size of the sphere.From the transformed sphere equation \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), you can determine the radius:
- Equate with the constant: In \((x-5)^2 + (y+1)^2 + (z+4)^2 = 51\), the term \(51\) is crucial.
- Calculate the radius: This number corresponds to \(r^2\), so by taking the square root, we find \(r = \sqrt{51}\), giving the radius of the sphere.
