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
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![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
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