For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve.
(a)
a circle with radius 3 and center (1, 3)

Incorrect: Your answer is incorrect.
(b)
a circle centered at the origin with radius 2

Let’s address each case and analyze the best equation form (polar or Cartesian) for the given curves:

### (a) A circle with radius 3 and center \((1, 3)\)

A circle that is not centered at the origin is more conveniently described using **Cartesian coordinates**. The general equation for a circle in Cartesian coordinates is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

For this circle:
- Center \((h, k) = (1, 3)\)
- Radius \(r = 3\)

Substituting these values into the equation gives:

\[
(x - 1)^2 + (y - 3)^2 = 9
\]

So, the Cartesian equation for this circle is:
\[
(x - 1)^2 + (y - 3)^2 = 9
\]

### (b) A circle centered at the origin with radius 2

When a circle is centered at the origin, it is often more easily described using **polar coordinates** because the radial distance \(r\) from the origin is constant. In polar coordinates, the general equation for a circle centered at the origin is:

\[
r = \text{constant}
\]

For this case, since the radius is 2, the equation in polar coordinates is simply:

\[
r = 2
\]

Alternatively, in Cartesian coordinates, the equation for a circle centered at the origin with radius \(r\) is:

\[
x^2 + y^2 = r^2
\]

So, the Cartesian equation would be:

\[
x^2 + y^2 = 4
\]

### Final Answers:
(a) Cartesian equation: \((x - 1)^2 + (y - 3)^2 = 9\)  
(b) Polar equation: \(r = 2\)

---

Do you have any questions or would you like further explanation? Here are five related questions for practice:

1. What is the polar equation for a circle of radius 5 centered at the origin?
2. How would you write the equation of a circle in Cartesian form if the center is \((-2, 4)\) and the radius is 6?
3. What is the polar equation for a vertical line passing through the origin?
4. How do you convert a polar equation like \(r = 3\) to Cartesian form?
5. What is the Cartesian equation for an ellipse with semi-major axis 5 and semi-minor axis 3 centered at the origin?

**Tip:** For curves centered at the origin, polar equations are often simpler than Cartesian ones.

For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve.
(a) a circle with radius 3 and center (1, 3)
(b) a circle centered at the origin with radius 2

Learn how to determine whether a curve is best described by a Cartesian or Polar equation. This tutorial covers the equations for circles, including examples with a circle centered at (1, 3) and a circle centered at the origin. Understand key concepts like Cartesian and Polar coordinates.

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