A sphere, Si is given by the equation: r? + y² + 22 – 2x – 4y + 8z + 17 = 0 (a) Write the equation for sphere Si in standard form (b) Sz is a sphere centered at (-3, -2, 1) and tangent to the xy plane. Find the distance from the surface of Si to the surface of S2.
A sphere, Si is given by the equation: r? + y² + 22 – 2x – 4y + 8z + 17 = 0 (a) Write the equation for sphere Si in standard form (b) Sz is a sphere centered at (-3, -2, 1) and tangent to the xy plane. Find the distance from the surface of Si to the surface of S2.
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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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa9efe910-fb5f-41b2-a82d-eda9e45aba44%2F34b3691c-9cac-4aef-a7a9-02f10f5723dc%2F3ps8gg_processed.png&w=3840&q=75)
Transcribed Image Text:### 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.
Now, write the equation in the standard form.
Thus, the standard form of the equation is
Step 2
(b)
For the sphere S2, the given center is (-3,-2,1). The tangent plane is xy plane in the plane.
So, the coordinate of the plane is .
Let the given the point p1 in the plane is (-3,-2,1) and the point p2, which lies in the xy plane, is (-3,-2,0).
Now, calculate the distance between both the points.
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