Chapter 9: Q. 81 (page 568)

The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
role="math" localid="1646746889027" $r = \frac{4}{1 - \cos θ}$

Short Answer

Expert verified

The equation is written using rectangular coordinates as $y^{2} = 8 (x + 2)$ .

Step by step solution

Step 1. Given Information

We are given an equation $r = \frac{4}{1 - \cos θ}$ .

We need to write the equation using rectangular coordinates.

Step 2. Concept Used

If P is a point with polar coordinates $(r, θ)$ , the rectangular coordinates $(x, y)$ of P are given by

$x = r \cos θ y = r \sin θ$

And it implies that $x^{2} + y^{2} = r^{2}$ .

Step 3. Convert into rectangular coordinates

In the given equation $r = \frac{4}{1 - \cos θ}$ , we multiply both sides by $1 - \cos θ$ and we get

$\begin{array}{rcl} r (1 - \cos θ) & = & \frac{4}{1 - \cos θ} (1 - \cos θ) \\ r - r \cos θ & = & 4 \end{array}$

Now using the relations we substitute $\sqrt{x^{2} + y^{2}}$ for $r$ and $x$ for $r \cos θ$ and we get

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

Step 4. Simplify the equation

Add $x$ to both sides of the equation and then square both sides and simplify.

role="math" $\begin{array}{rcl} \sqrt{x^{2} + y^{2}} - x + x & = & 4 + x \\ \sqrt{x^{2} + y^{2}} & = & 4 + x \\ {(\sqrt{x^{2} + y^{2}})}^{2} & = & {(4 + x)}^{2} \\ x^{2} + y^{2} & = & 16 + 8 x + x^{2} \end{array}$

Subtract $x^{2}$ from both sides.

$\begin{array}{rcl} x^{2} + y^{2} - x^{2} & = & 16 + 8 x + x^{2} - x^{2} \\ y^{2} & = & 8 x + 16 \\ y^{2} & = & 8 (x + 2) \end{array}$

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The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
role="math" localid="1646746889027" $r = \frac{4}{1 - \cos θ}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept Used

Step 3. Convert into rectangular coordinates

Step 4. Simplify the equation

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The letters r and θrepresent polar coordinates. Write each equation using rectangular coordinates x,y.role="math" localid="1646746889027" r=41-cosθ

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Concept Used

Step 3. Convert into rectangular coordinates

Step 4. Simplify the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

The letters r and $θ$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$ .
role="math" localid="1646746889027" $r = \frac{4}{1 - \cos θ}$