f(x, y) = x² + y² - 10x - 20y Expand (x - 5)² + (y - 10)² Show that for k> -125, the level set f(x,y) = k is a circle Describe the level set f(x, y) -125 ==
f(x, y) = x² + y² - 10x - 20y Expand (x - 5)² + (y - 10)² Show that for k> -125, the level set f(x,y) = k is a circle Describe the level set f(x, y) -125 ==
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The text presents a problem involving level sets and the expansion of perfect squares. Here is the transcription and explanation:
---
**Problem:**
Given the function:
\[ f(x, y) = x^2 + y^2 - 10x - 20y \]
1. **Expand \((x - 5)^2 + (y - 10)^2\)**
2. **Show that for \(k > -125\), the level set \(f(x, y) = k\) is a circle.**
3. **Describe the level set \(f(x, y) = -125\).**
---
**Analysis and Solution:**
1. **Expansion of \((x - 5)^2 + (y - 10)^2\):**
\[
(x - 5)^2 = x^2 - 10x + 25
\]
\[
(y - 10)^2 = y^2 - 20y + 100
\]
Therefore,
\[
(x - 5)^2 + (y - 10)^2 = x^2 - 10x + 25 + y^2 - 20y + 100
\]
Simplifying,
\[
= x^2 + y^2 - 10x - 20y + 125
\]
2. **Show that for \(k > -125\), the level set \(f(x, y) = k\) is a circle:**
From the expansion in step 1,
\[
f(x, y) = (x - 5)^2 + (y - 10)^2 - 125
\]
Setting this equal to \(k\),
\[
(x - 5)^2 + (y - 10)^2 = k + 125
\]
This is the equation of a circle centered at \((5, 10)\) with radius \(\sqrt{k + 125}\).
3. **Describe the level set \(f(x, y) = -125\):**
Substituting \(-125\) into the equation,
\[
(x - 5)^2 + (y - 10)^2 = 0
\]
This implies](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1467f162-7185-45ab-925a-2589d3c8cce7%2F951dcbe2-9f1c-4eec-b45a-26318a355020%2F848iqif_processed.png&w=3840&q=75)
Transcribed Image Text:The text presents a problem involving level sets and the expansion of perfect squares. Here is the transcription and explanation:
---
**Problem:**
Given the function:
\[ f(x, y) = x^2 + y^2 - 10x - 20y \]
1. **Expand \((x - 5)^2 + (y - 10)^2\)**
2. **Show that for \(k > -125\), the level set \(f(x, y) = k\) is a circle.**
3. **Describe the level set \(f(x, y) = -125\).**
---
**Analysis and Solution:**
1. **Expansion of \((x - 5)^2 + (y - 10)^2\):**
\[
(x - 5)^2 = x^2 - 10x + 25
\]
\[
(y - 10)^2 = y^2 - 20y + 100
\]
Therefore,
\[
(x - 5)^2 + (y - 10)^2 = x^2 - 10x + 25 + y^2 - 20y + 100
\]
Simplifying,
\[
= x^2 + y^2 - 10x - 20y + 125
\]
2. **Show that for \(k > -125\), the level set \(f(x, y) = k\) is a circle:**
From the expansion in step 1,
\[
f(x, y) = (x - 5)^2 + (y - 10)^2 - 125
\]
Setting this equal to \(k\),
\[
(x - 5)^2 + (y - 10)^2 = k + 125
\]
This is the equation of a circle centered at \((5, 10)\) with radius \(\sqrt{k + 125}\).
3. **Describe the level set \(f(x, y) = -125\):**
Substituting \(-125\) into the equation,
\[
(x - 5)^2 + (y - 10)^2 = 0
\]
This implies
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
