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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 28. (page 683)

Convert each polar equation into a rectangular equation.
$r = \frac{10}{5 + 4 \cos (胃)}$

Short Answer

Expert verified

The rectangular equation is

$9 x^{2} + 25 y^{2} + 80 x = 100$

Step by step solution

01

Step 1. Given information

The polar equation is

$r = \frac{10}{5 + 4 \cos (胃)}$

02

Step 2. Convert into the polar equation

The polar equation is

$r = \frac{10}{5 + 4 \cos (胃)}$

Multiply both sides by the denominator

$r 脳 (5 + 4 \cos (胃)) = \frac{10}{5 + 4 \cos (胃)} 脳 5 + 4 \cos (胃) r 脳 (5 + 4 \cos (胃)) = 10 5 r + 4 r \cos (胃) = 10$

Apply conversion equation formula

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

$5 \sqrt{x^{2} + y^{2}} + 4 x = 10$

Subtract both sides by $4 x$

$5 \sqrt{x^{2} + y^{2}} + 4 x - 4 x = 10 - 4 x 5 \sqrt{x^{2} + y^{2}} = 10 - 4 x$

Take square on both sides

${(5 \sqrt{x^{2} + y^{2}})}^{2} = {(10 - 4 x)}^{2} 25 (x^{2} + y^{2}) = 100 - 80 x + 16 x^{2} 25 x^{2} + 25 y^{2} = 100 - 80 x + 16 x^{2} 25 x^{2} + 25 y^{2} + 80 x - 16 x^{2} = 100 9 x^{2} + 25 y^{2} + 80 x = 100$

So, the rectangular equation is

$9 x^{2} + 25 y^{2} + 80 x = 100$

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