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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 35. (page 683)

Convert each polar equation into a rectangular equation.
$r = \frac{6 s e c (胃)}{2 s e c (胃) - 1}$

Short Answer

Expert verified

The rectangular equation is

$3 x^{2} + 4 y^{2} - 12 x = 36$

Step by step solution

01

Step 1. Given information

The polar equation is

$r = \frac{6 s e c (胃)}{2 s e c (胃) - 1}$

02

Step 2. Convert into the rectangular equation

$x$ The polar equation is

$r = \frac{6 s e c (胃)}{2 s e c (胃) - 1}$

Multiply top and bottom by $\cos (胃)$

$r = \frac{6 s e c (胃) 脳 \cos (胃)}{(2 s e c (胃) - 1) 脳 \cos (胃)} r = \frac{6 脳 \frac{1}{\cos (胃)} 脳 \cos (胃)}{(2 \frac{1}{\cos (胃)} 脳 \cos (胃) - \cos (胃))} r = \frac{6}{2 - \cos (胃)}$

Multiply denominator on both sides

$r 脳 (2 - \cos (胃)) = \frac{6}{2 - \cos (胃)} 脳 2 - \cos (胃) r (2 - \cos (胃)) = 6 2 r - r \cos (胃) = 6$

Apply the conversion formula

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

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

Add both sides by $x$

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

Take square on both sides and then simplify

${(2 \sqrt{x^{2} + y^{2}})}^{2} = {(6 + x)}^{2} 4 (x^{2} + y^{2}) = 36 + x^{2} + 12 x 4 x^{2} + 4 y^{2} = 36 + x^{2} + 12 x 4 x^{2} + 4 y^{2} - x^{2} - 12 x = 36 3 x^{2} + 4 y^{2} - 12 x = 36$

So, the rectangular equation is

$3 x^{2} + 4 y^{2} - 12 x = 36$

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