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Chapter 12: Problem 29

Give a geometric description of the following sets of points. $$(x-1)^{2}+y^{2}+z^{2}-9=0$$

Short Answer

Expert verified
Question: Identify the geometric shape, center, and radius represented by the given equation: $$(x-1)^{2}+y^{2}+z^{2}-9=0$$ Answer: The given equation represents a sphere with a center at (1, 0, 0) and a radius of 3.

Step by step solution

01

Rewrite the equation in a more recognizable form

Add 9 to both sides of the equation to obtain: $$(x-1)^{2}+y^{2}+z^{2} = 9$$ The equation now looks similar to the equation of a sphere, which is given by: $$(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}$$ where \((a, b, c)\) are the coordinates of the center, and \(r\) is the radius.
02

Identify the center

We can identify the center by looking at the coefficients of the \((x-1)\), \(y\), and \(z\) terms. Comparing the given equation with the general equation for a sphere, we see that the center is \((1, 0, 0)\).
03

Identify the radius

The radius is given by the square root of the constant value on the right side of the equation, which in this case is \(9\). Thus, the radius is \(3\).
04

Write the geometric description

The geometric description of the given set of points is a sphere with a center at \((1, 0, 0)\) and a radius of \(3\).

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