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
![**Convert the rectangular equation to polar form.**
\[ (x^2 + y^2)^2 = 7 (x^2 - y^2) \]
**Solution:**
_Rectangular to Polar Coordinates Conversion:_
To convert the given rectangular equation \( (x^2 + y^2)^2 = 7 (x^2 - y^2) \) to polar form, we need to use the relationships between rectangular coordinates \((x, y)\) and polar coordinates \((r, \theta)\):
\[ x = r \cos (\theta) \]
\[ y = r \sin (\theta) \]
From these, it follows that:
\[ x^2 + y^2 = r^2 \]
\[ x^2 - y^2 = r^2 \cos (2\theta) \]
_Substituting these into the given equation:_
Given equation:
\[ (x^2 + y^2)^2 = 7 (x^2 - y^2) \]
Substitute \( x^2 + y^2 = r^2 \) and \( x^2 - y^2 = r^2 \cos (2\theta) \):
\[ (r^2)^2 = 7 (r^2 \cos (2\theta)) \]
Simplify the equation:
\[ r^4 = 7r^2 \cos(2\theta) \]
Since \( r \neq 0 \), divide both sides by \( r^2 \):
\[ r^2 = 7 \cos (2\theta) \]
Thus, the polar form of the given equation is:
\[ r^2 = 7 \cos (2\theta) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc51a27f7-e004-47b6-8023-544dc66e83a5%2F95896730-3a71-4964-8936-887d2e1f323f%2Fy0vs1xb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Convert the rectangular equation to polar form.**
\[ (x^2 + y^2)^2 = 7 (x^2 - y^2) \]
**Solution:**
_Rectangular to Polar Coordinates Conversion:_
To convert the given rectangular equation \( (x^2 + y^2)^2 = 7 (x^2 - y^2) \) to polar form, we need to use the relationships between rectangular coordinates \((x, y)\) and polar coordinates \((r, \theta)\):
\[ x = r \cos (\theta) \]
\[ y = r \sin (\theta) \]
From these, it follows that:
\[ x^2 + y^2 = r^2 \]
\[ x^2 - y^2 = r^2 \cos (2\theta) \]
_Substituting these into the given equation:_
Given equation:
\[ (x^2 + y^2)^2 = 7 (x^2 - y^2) \]
Substitute \( x^2 + y^2 = r^2 \) and \( x^2 - y^2 = r^2 \cos (2\theta) \):
\[ (r^2)^2 = 7 (r^2 \cos (2\theta)) \]
Simplify the equation:
\[ r^4 = 7r^2 \cos(2\theta) \]
Since \( r \neq 0 \), divide both sides by \( r^2 \):
\[ r^2 = 7 \cos (2\theta) \]
Thus, the polar form of the given equation is:
\[ r^2 = 7 \cos (2\theta) \]
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