https://file.aiquickdraw.com/mathbot/file/1731729222848whqtspu1.jpg

To convert the given polar equation \( r = \frac{3}{1 - \cos\theta} \) into rectangular coordinates (\(x, y\)), we will use the relationships between polar and rectangular coordinates:

1. \( x = r \cos\theta \)
2. \( y = r \sin\theta \)
3. \( r^2 = x^2 + y^2 \)
4. \( \cos\theta = \frac{x}{r} \)

### Step-by-Step Conversion:

1. Start with the polar equation:
   \[
   r = \frac{3}{1 - \cos\theta}
   \]

2. Multiply through by \( 1 - \cos\theta \) to eliminate the denominator:
   \[
   r(1 - \cos\theta) = 3
   \]

3. Expand:
   \[
   r - r\cos\theta = 3
   \]

4. Substitute \( r\cos\theta = x \):
   \[
   r - x = 3
   \]

5. Solve for \( r \):
   \[
   r = x + 3
   \]

6. Substitute \( r^2 = x^2 + y^2 \):
   \[
   (x + 3)^2 = x^2 + y^2
   \]

7. Expand \( (x + 3)^2 \):
   \[
   x^2 + 6x + 9 = x^2 + y^2
   \]

8. Simplify by canceling \( x^2 \) from both sides:
   \[
   6x + 9 = y^2
   \]

9. Rearrange to get the general form:
   \[
   0 = y^2 - 6x - 9
   \]

### Final Answer:
The rectangular form of the equation is:
\[
0 = y^2 - 6x - 9
\]

Would you like a more detailed explanation of any step or have questions? 

---

Here are 5 related questions for further exploration:
1. How do we convert other polar equations, such as \( r = 2\sin\theta \), to rectangular form?
2. What type of conic section does this rectangular equation represent?
3. Can the equation be rewritten in standard form for a parabola?
4. How would the equation change if \( \cos\theta \) was replaced with \( \sin\theta \)?
5. How do the transformations \( x = r\cos\theta \) and \( y = r\sin\theta \) work geometrically?

**Tip:** Always check if the final rectangular equation corresponds to a familiar geometric shape (circle, parabola, ellipse, or hyperbola) for a better understanding of the problem!

Write the given polar equation r = 3 / (1 - cos(θ)) using rectangular coordinates (x, y).

Learn how to convert the polar equation r = 3 / (1 - cos(θ)) into rectangular coordinates (x, y). This step-by-step solution covers key formulas and mathematical concepts, including the relationship between polar and rectangular coordinate systems. Perfect for high school students studying conic sections and advanced coordinate geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation