Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 8 - Section 8.1 - Polar Coordinates - 8.1 Exercises - Page 593: 60

Answer

$9(x^{2}+y^{2})=(x^{2}+y^{2}+3x)^{2}$

Work Step by Step

Formulas relating rectangular $(x,y)$ and polar $(r,\theta)$ coordinates: $ x=r\cos\theta$ and $ y=r\sin\theta$ $r^{2}=x^{2}+y^{2}$ and $\displaystyle \tan\theta=\frac{y}{x}$. --------- $ r=3-3\cos\theta$ ... multiply both sides with r, $ r^{2}=3r-3r\cos\theta$ ... substitute $r^{2}$ and $ r\cos\theta$ using the formulas $x^{2}+y^{2}=3r-3x\qquad /+3x$ ... isolate r on one side, $ 3r=x^{2}+y^{2}+3x\qquad$... square both sides, $9r^{2}=(x^{2}+y^{2}+3x)^{2}\qquad $... substitute $r^{2}$ $9(x^{2}+y^{2})=(x^{2}+y^{2}+3x)^{2}$, ... an equation in rectangular coordinates.
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