### Problem StatementA sphere, $ S_1 $, is given by the equation: \[ x^2 + y^2 + z^2 - 2x - 4y + 8z + 17 = 0 \](a) Write the equation for sphere $ S_1 $ in standard form.(b) $ S_2 $ is a sphere centered at $ (-3, -2, 1) $ and tangent to the xy plane. Find the distance from the surface of $ S_1 $ to the surface of $ S_2 $.### Explanation- To convert the equation of sphere $ S_1 $ to standard form, you will complete the square for each variable $ x $, $ y $, and $ z $.- For sphere $ S_2 $, since it is tangent to the xy-plane, consider its distance from the center to the plane to find its radius.- The distance between the surfaces involves calculating the distance between the centers and subtracting the radii of both spheres.

Answered: A sphere, Si is given by the equation:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

### Problem Statement

A sphere, $ S_1 $, is given by the equation:

\[ x^2 + y^2 + z^2 - 2x - 4y + 8z + 17 = 0 \]

(a) Write the equation for sphere $ S_1 $ in standard form.

(b) $ S_2 $ is a sphere centered at $ (-3, -2, 1) $ and tangent to the xy plane. Find the distance from the surface of $ S_1 $ to the surface of $ S_2 $.

### Explanation

- To convert the equation of sphere $ S_1 $ to standard form, you will complete the square for each variable $ x $, $ y $, and $ z $.
- For sphere $ S_2 $, since it is tangent to the xy-plane, consider its distance from the center to the plane to find its radius.
- The distance between the surfaces involves calculating the distance between the centers and subtracting the radii of both spheres.

Expert Solution

Step 1

(a)

Consider the given information.

$x^{2} + y^{2} + z^{2} - 2 x - 4 y + 8 z + 17 = 0$

Now, write the equation in the standard form.

$\begin{array}{rcl} x^{2} - 2 x + y^{2} - 4 y + z^{2} + 8 z + 17 & = & 0 \\ x^{2} - 2 \cdot x \cdot 1 + 1^{2} + y^{2} - 2 \cdot y \cdot 2 + 2^{2} + z^{2} + 2 \cdot z \cdot 4 + 4^{2} & = & - 17 + 1^{2} + 2^{2} + 4^{2} \\ {(x - 1)}^{2} + {(y - 2)}^{2} + {(z + 4)}^{2} & = & - 17 + 1 + 4 + 16 \\ {(x - 1)}^{2} + {(y - 2)}^{2} + {(z + 4)}^{2} & = & 4 \end{array}$