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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 32. (page 683)

Convert each polar equation into a rectangular equation.
$r = \frac{8}{2 + 4 \cos (胃)}$

Short Answer

Expert verified

The rectangular equation is

$y^{2} - 3 x^{2} + 16 x = 16$

Step by step solution

01

Step 1. Given information

The polar equation is

$r = \frac{8}{2 + 4 \cos (胃)}$

02

Step 2. Convert into the rectangular equation

The polar equation is

$r = \frac{8}{2 + 4 \cos (胃)}$

Multiply both sides by the denominator

$r 脳 (2 + 4 \cos (胃)) = \frac{8}{2 + 4 \cos (胃)} 脳 2 + 4 \cos (胃) r (2 + 4 \cos (胃)) = 8 2 r (1 + 2 \cos (胃)) = 8 r (1 + 2 \cos (胃)) = 4 r + 2 r \cos (胃) = 4$

Apply the conversion formula

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

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

Subtract both sides by $2 x$

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

Take square on both sides and simplify it

${(\sqrt{x^{2} + y^{2}})}^{2} = {(4 - 2 x)}^{2} x^{2} + y^{2} = 16 + 4 x^{2} - 16 x x^{2} + y^{2} - 4 x^{2} + 16 x = 16 y^{2} - 3 x^{2} + 16 x = 16$

So, the rectangular equation is $y^{2} - 3 x^{2} + 16 x = 16$

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Most popular questions from this chapter

Find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.

$x^{2} = - 4 y$

Find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.

$y^{2} = 8 x$

2. To complete the square of $x^{2} - 4 x$ ,add _______.

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