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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 75 (page 622)

In Problems $71 鈥� 76$ , find the radius and center of each sphere.
$2 x^{2} + 2 y^{2} + 2 z^{2} - 8 x + 4 z = - 1$

Short Answer

Expert verified

The center is $(2, 0, - 1)$ and radius is $\frac{3 \sqrt{2}}{2}$

Step by step solution

01

Step 1. Given information

An equation is given as $2 x^{2} + 2 y^{2} + 2 z^{2} - 8 x + 4 z = - 1$

02

Step 2. Finding center and radius

First convert equation in the form as

$r^{2} = {(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}$

Where ris radius and center is $P_{0} = (x_{0}, y_{0}, z_{0})$

Simplify as,

$2 x^{2} + 2 y^{2} + 2 z^{2} - 8 x + 4 z = - 1 x^{2} + y^{2} + z^{2} - 4 x + 2 z = - \frac{1}{2} (x^{2} - 4 x) + y^{2} + (z^{2} + 2 z) = - \frac{1}{2} (x^{2} - 4 x + 4) + y^{2} + (z^{2} + 2 z + 1) = - \frac{1}{2} + 4 + 1 (x - 2)^{2} + y^{2} + (z + 1)^{2} = \frac{9}{2}$

Therefore the center is $(2, 0, - 1)$ and radius is $\frac{3 \sqrt{2}}{2}$

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