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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 10: Q 29. (page 683)

Convert each polar equation into a rectangular equation.
$r = \frac{9}{3 - 6 \cos (胃)}$

Short Answer

Expert verified

The rectangular equation is

$9 y^{2} - 27 x^{2} - 108 x = 81$

Step by step solution

01

Step 1. Given information

The polar equation is

$r = \frac{9}{3 - 6 \cos (胃)}$

02

Step 2. Convert into the rectangular equation

The polar equation is

$r = \frac{9}{3 - 6 \cos (胃)}$

Multiply denominator on both sides

r 脳 (3 - 6 \cos (胃)) = \frac{9}{3 - 6 \cos (胃)} 脳 3 - 6 \cos (胃) r 脳 (3 - 6 \cos (胃)) = 9 3 r - 6 r \cos (胃) = 9

Apply conversion equation formula

$r = \sqrt{x^{2} + y^{2}}, x = r \cos (胃)$

$3 \sqrt{x^{2} + y^{2}} - 6 x = 9$

Add both sides by $6 x$

$3 \sqrt{x^{2} + y^{2}} - 6 x + 6 x = 9 + 6 x 3 \sqrt{x^{2} + y^{2}} = 9 + 6 x$

Take square on both sides

${(3 \sqrt{x^{2} + y^{2}})}^{2} = {(9 + 6 x)}^{2} 9 (x^{2} + y^{2}) = 81 + 108 x + 36 x^{2} 9 x^{2} + 9 y^{2} = 81 + 108 x + 36 x^{2} 9 x^{2} + 9 y^{2} - 108 x - 36 x^{2} = 81 9 y^{2} - 27 x^{2} - 108 x = 81$

So, the rectangular equation is

$9 y^{2} - 27 x^{2} - 108 x = 81$

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