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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 30. (page 683)

Convert each polar equation into a rectangular equation.
$r = \frac{12}{4 + 8 \sin (胃)}$

Short Answer

Expert verified

The rectangular equation is

$x^{2} - 3 y^{2} + 12 y = 9$

Step by step solution

01

Step 1. Given information

The polar equation is

$r = \frac{12}{4 + 8 \sin (胃)}$

02

Step 2. Convert into the polar equation.

The polar equation is

$r = \frac{12}{4 + 8 \sin (胃)}$

Multiply denominator on both sides

$r 脳 (4 + 8 \sin (胃)) = \frac{12}{4 + 8 \sin (胃)} 脳 (4 + 8 \sin (胃)) r (4 + 8 \sin (胃)) = 12 4 r + 8 r \sin (胃) = 12$

Divide both sides by $4$

role="math" localid="1647259342478" $r + 2 r \sin (胃) = 3$

Apply conversion equation formula

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

role="math" localid="1647259639691" $\sqrt{x^{2} + y^{2}} + 2 y = 3$

Subtract both sides by y

role="math" localid="1647259661067" $\sqrt{x^{2} + y^{2}} + 2 y - 2 y = 3 - 2 y \sqrt{x^{2} + y^{2}} = 3 - 2 y$

Take square on both sides

${(\sqrt{x^{2} + y^{2}})}^{2} = {(3 - 2 y)}^{2} x^{2} + y^{2} = 9 + 4 y^{2} - 12 y x^{2} + y^{2} - 4 y^{2} + 12 y = 9 x^{2} - 3 y^{2} + 12 y = 9$

So, the rectangular equation is $x^{2} - 3 y^{2} + 12 y = 9$

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