Answer:
Let's analyze each curve and determine whether it would be more easily described by a polar equation or a Cartesian equation:
(a) A circle with radius 3 and center (3,2):
In this case, the center of the circle is not at the origin, but at (3, 2). To describe this circle using polar coordinates, it would be more complicated, as polar coordinates naturally center around the origin. Therefore, it would be more convenient to use a Cartesian equation to describe this circle.
The Cartesian equation of a circle with radius \(r\) and center \((h, k)\) is given by:
\((x - h)^2 + (y - k)^2 = r^2\)
In this case, \(r = 3\), \(h = 3\), and \(k = 2\), so the Cartesian equation for this circle is:
\((x - 3)^2 + (y - 2)^2 = 3^2\)
(b) A circle centered at the origin with radius 1:
For a circle centered at the origin, it is often simpler to use polar coordinates because the equation for a circle in polar coordinates is quite straightforward:
\(r = \text{constant}\)
In this case, the constant is the radius, which is 1. So, the polar equation for this circle is:
\(r = 1\)
This equation directly states that the radius (\(r\)) is always equal to 1, making it a straightforward representation of a circle centered at the origin.
Explanation: