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The given polar equation is:

\[
r = \frac{1}{1 - 2 \cos \theta}
\]

To understand the shape of the curve represented by this polar equation, let's analyze it step-by-step.

### Step 1: Identify the Type of Conic Section

The general form of a polar equation for a conic section is:

\[
r = \frac{l}{1 + e \cos \theta} \quad \text{or} \quad r = \frac{l}{1 + e \sin \theta}
\]

where:
- \( l \) is the semi-latus rectum,
- \( e \) is the eccentricity.

### Step 2: Compare with the General Form

The given equation is:

\[
r = \frac{1}{1 - 2 \cos \theta}
\]

We can rewrite it as:

\[
r = \frac{1}{1 + (-2) \cos \theta}
\]

Here, \( e = 2 \) (eccentricity) and \( l = 1 \) (semi-latus rectum).

### Step 3: Determine the Type of Conic Section

- If \( e = 1 \), the conic is a parabola.
- If \( e < 1 \), the conic is an ellipse.
- If \( e > 1 \), the conic is a hyperbola.

Since \( e = 2 > 1 \), the conic section represented by the given equation is a **hyperbola**.

### Conclusion

The graph of the polar equation \( r = \frac{1}{1 - 2 \cos \theta} \) is a **hyperbola**.

Would you like more details or have any questions?

### Related Questions:
1. How do we graph polar equations of conic sections?
2. What are the properties of hyperbolas in polar coordinates?
3. How can we convert polar equations to Cartesian coordinates?
4. What is the significance of the eccentricity in conic sections?
5. How can we find the asymptotes of a hyperbola in polar form?

### Tip:
To sketch a polar curve, it helps to convert a few polar coordinates to Cartesian coordinates to get a better understanding of the shape.

Given the polar equation r = 1 / (1 - 2cosθ), determine the shape of the curve.

Explore how to determine the shape of the curve represented by the polar equation r = 1 / (1 - 2cosθ). This problem delves into the classification of conic sections based on eccentricity, making it ideal for high school students in Grades 11-12.

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