Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem:**
Find the polar equation of the line \( y = 10x + 9 \) in terms of \( r \) and \( \theta \).
(Use symbolic notation and fractions where needed.)
**Solution:**
To convert the given Cartesian equation \( y = 10x + 9 \) into polar coordinates, we use the relationships:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
Substitute these into the equation:
\[ r \sin \theta = 10(r \cos \theta) + 9 \]
Simplify to solve for \( r \):
\[ r \sin \theta = 10r \cos \theta + 9 \]
\[ r (\sin \theta - 10 \cos \theta) = 9 \]
\[ r = \frac{9}{\sin \theta - 10 \cos \theta} \]
Therefore, the polar equation is:
\[ r = \frac{9}{\sin \theta - 10 \cos \theta} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9b629bd-7b7c-4b52-85cc-46a95271ddc8%2Fe97bd71c-a6d0-44aa-ad9d-b4cab39fc50f%2Fcnjhtj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find the polar equation of the line \( y = 10x + 9 \) in terms of \( r \) and \( \theta \).
(Use symbolic notation and fractions where needed.)
**Solution:**
To convert the given Cartesian equation \( y = 10x + 9 \) into polar coordinates, we use the relationships:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
Substitute these into the equation:
\[ r \sin \theta = 10(r \cos \theta) + 9 \]
Simplify to solve for \( r \):
\[ r \sin \theta = 10r \cos \theta + 9 \]
\[ r (\sin \theta - 10 \cos \theta) = 9 \]
\[ r = \frac{9}{\sin \theta - 10 \cos \theta} \]
Therefore, the polar equation is:
\[ r = \frac{9}{\sin \theta - 10 \cos \theta} \]
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