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: $r = 2$

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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.