Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Problem Statement:
Determine the coordinates of the center and the radius of the circle given by the equation \(r \csc(\theta) = -4\). Convert the polar equation to a rectangular equation if necessary.
(Use symbolic notation and fractions where needed. Give your answer as a comma-separated list of a point's coordinates in the form \((\ast, \ast)\))
---
**Solution:**
#### The coordinates of the center:
\[
(0, -2)
\]
#### Radius:
\[
2
\]
#### The excluded points:
\[
+ k\pi \text{ for some integer } k
\]
---
### Explanation:
1. **Polar to Rectangular Conversion:**
- The given equation is in polar form: \( r \csc(\theta) = -4 \).
- We can rewrite \( \csc(\theta) \) as \( \frac{1}{\sin(\theta)} \), therefore, the equation becomes:
\[
r \cdot \frac{1}{\sin(\theta)} = -4
\]
\[
r = -4 \sin(\theta)
\]
2. **Identify Coordinates and Radius:**
- In polar form, \( r \) represents the distance from a point to the origin and \( \theta \) represents the angle from the positive x-axis.
- To find the center and radius of this circle, recognize that this equation describes a circle with its center shifted.
3. **Rectangular Coordinates:**
- Converting the polar coordinates to rectangular, use \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
- Substituting \( r = -4 \sin(\theta) \):
\[
x = -4 \sin(\theta) \cos(\theta)
\]
\[
y = -4 \sin^2(\theta)
\]
- Use the equation of the circle in rectangular form \( (x - h)^2 + (y - k)^2 = R^2 \) to identify the center (\(h, k\)) and radius \(R\).
4. **Excluded Points:**
- The sine function is zero whenever \( \theta = k \pi \) for any integer \( k \). Therefore, points where \( \theta \) is an](https://content.bartleby.com/qna-images/question/609cd648-ee60-4e65-94f5-e2f0e2671842/d77a21a7-8963-4bd1-b764-ce97656c1093/kku4esb_processed.jpeg)
Transcribed Image Text:### Problem Statement:
Determine the coordinates of the center and the radius of the circle given by the equation \(r \csc(\theta) = -4\). Convert the polar equation to a rectangular equation if necessary.
(Use symbolic notation and fractions where needed. Give your answer as a comma-separated list of a point's coordinates in the form \((\ast, \ast)\))
---
**Solution:**
#### The coordinates of the center:
\[
(0, -2)
\]
#### Radius:
\[
2
\]
#### The excluded points:
\[
+ k\pi \text{ for some integer } k
\]
---
### Explanation:
1. **Polar to Rectangular Conversion:**
- The given equation is in polar form: \( r \csc(\theta) = -4 \).
- We can rewrite \( \csc(\theta) \) as \( \frac{1}{\sin(\theta)} \), therefore, the equation becomes:
\[
r \cdot \frac{1}{\sin(\theta)} = -4
\]
\[
r = -4 \sin(\theta)
\]
2. **Identify Coordinates and Radius:**
- In polar form, \( r \) represents the distance from a point to the origin and \( \theta \) represents the angle from the positive x-axis.
- To find the center and radius of this circle, recognize that this equation describes a circle with its center shifted.
3. **Rectangular Coordinates:**
- Converting the polar coordinates to rectangular, use \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
- Substituting \( r = -4 \sin(\theta) \):
\[
x = -4 \sin(\theta) \cos(\theta)
\]
\[
y = -4 \sin^2(\theta)
\]
- Use the equation of the circle in rectangular form \( (x - h)^2 + (y - k)^2 = R^2 \) to identify the center (\(h, k\)) and radius \(R\).
4. **Excluded Points:**
- The sine function is zero whenever \( \theta = k \pi \) for any integer \( k \). Therefore, points where \( \theta \) is an
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