Chapter 10: Q 36. (page 683)

Convert each polar equation into a rectangular equation.
$r = \frac{3 c s c (θ)}{c s c (θ) - 1}$

Short Answer

Expert verified

The rectangular equation is

$x^{2} - 6 y = 9$

Step by step solution

Step 1. Given information

The polar equation is

$r = \frac{3 c s c (θ)}{c s c (θ) - 1}$

Step 2. Convert into the rectangular equation

The polar equation is

$r = \frac{3 c s c (θ)}{c s c (θ) - 1}$

Multiply top and bottom by $\sin (θ)$

$r = \frac{3 c s c (θ) \times \sin (θ)}{c s c (θ) \times \sin (θ) - 1 \times \sin (θ)} r = \frac{3 \frac{1}{\sin (θ)} \times \sin (θ)}{\frac{1}{\sin (θ)} \times \sin (θ) - 1 \times \sin (θ)} r = \frac{3}{1 - \sin (θ)}$

Multiply denominator on both sides

$r \times (1 - \sin (θ)) = \frac{3}{1 - \sin (θ)} \times 1 - \sin (θ) r \times (1 - \sin (θ)) = 3 r - r \sin (θ) = 3$

Apply the conversion formula

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

$\sqrt{x^{2} + y^{2}} - y = 3$

Add both sides by $y$

$\sqrt{x^{2} + y^{2}} - y + y = 3 + y \sqrt{x^{2} + y^{2}} = 3 + y$

Take square on both sides and simplify it

${(\sqrt{x^{2} + y^{2}})}^{2} = {(3 + y)}^{2} x^{2} + y^{2} = 9 + 6 y + y^{2} x^{2} + y^{2} - 6 y - y^{2} = 9 x^{2} - 6 y = 9$

So, the rectangular equation is

$x^{2} - 6 y = 9$

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Convert each polar equation into a rectangular equation.
$r = \frac{3 c s c (θ)}{c s c (θ) - 1}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Convert into the rectangular equation

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Convert each polar equation into a rectangular equation.r=3csc(θ)csc(θ)-1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Convert into the rectangular equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Applied Mathematics

Mechanics Maths

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

Convert each polar equation into a rectangular equation.
$r = \frac{3 c s c (θ)}{c s c (θ) - 1}$