Chapter 10: Q 30. (page 683)

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

Short Answer

Expert verified

The rectangular equation is

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

Step by step solution

Step 1. Given information

The polar equation is

$r = \frac{12}{4 + 8 \sin (θ)}$

Step 2. Convert into the polar equation.

The polar equation is

$r = \frac{12}{4 + 8 \sin (θ)}$

Multiply denominator on both sides

$r \times (4 + 8 \sin (θ)) = \frac{12}{4 + 8 \sin (θ)} \times (4 + 8 \sin (θ)) r (4 + 8 \sin (θ)) = 12 4 r + 8 r \sin (θ) = 12$

Divide both sides by $4$

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

Apply conversion equation formula

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

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Convert into the polar equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Mechanics Maths

Probability and Statistics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

Convert each polar equation into a rectangular equation.r=124+8sin(θ)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Convert into the polar equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Mechanics Maths

Probability and Statistics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

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