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To find the Cartesian equation for the curve given by \( r = 8 \sec(\theta) \), we can proceed with the following steps:

### Step 1: Convert \( \sec(\theta) \) to Cartesian form
We know that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Thus, the given equation becomes:
\[
r = \frac{8}{\cos(\theta)}
\]

### Step 2: Use the polar-to-Cartesian conversion formulas
The polar-to-Cartesian coordinate conversions are:
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- \( r^2 = x^2 + y^2 \)

From the equation \( r = \frac{8}{\cos(\theta)} \), multiply both sides by \( \cos(\theta) \):
\[
r \cos(\theta) = 8
\]
Using the conversion \( r \cos(\theta) = x \), we get:
\[
x = 8
\]

### Final Cartesian Equation:
Thus, the Cartesian equation of the curve is simply:
\[
x = 8
\]

### Step 3: Identify the curve
The equation \( x = 8 \) represents a **vertical line** at \( x = 8 \). Therefore, the correct answer for the type of curve is:
- **Line**

### Summary:
- The Cartesian equation is \( x = 8 \).
- The curve is a **line**.

Do you want more details or have any questions?

Here are some additional related questions to expand the topic:
1. What are the general forms of polar-to-Cartesian conversion formulas?
2. How can the equation \( r = a \sec(\theta) \) change if we modify the constant \( a \)?
3. What would the graph look like if we changed the trigonometric function from \( \sec(\theta) \) to \( \csc(\theta) \)?
4. How do polar equations like this represent shapes such as circles or hyperbolas?
5. Can a polar equation describe more complex curves like limacons or spirals?

**Tip**: When converting polar equations to Cartesian form, always check for symmetries or simple patterns in the polar function, which can lead to straightforward equations like lines or circles.

Find the Cartesian equation for the curve given by r = 8 sec(θ). Identify the type of curve.

Learn how to convert the polar equation r = 8 sec(θ) into Cartesian form, yielding x = 8. Explore the solution step-by-step and identify that the resulting curve is a vertical line.

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