Question

For constants \(a\) and \(b\), describe the graphs of the equations \(r=a\) and \(\theta=b\) in polar coordinates.

Short answer

The graph of the polar equation \(r=a\) is a circle with radius \(a\) centered at the origin. The graph of the polar equation \(\theta=b\) is a ray from the origin that makes an angle \(b\) with the positive x-axis.

Step-by-step solution

Graph of \(r=a\)

The equation \(r=a\) describes a circle with radius \(a\). It's graphed in polar coordinates by marking the value of \(a\) on the radial axis and drawing a circle with radius \(a\) centered at the origin. If \(a\) is positive, the graph is the circle of radius \(a\) centered at the origin. If \(a\) is negative, the graph is still the same circle, but plotted in the opposite direction.

Graph of \(\theta = b\)

The equation \(\theta=b\) describes a ray from the origin making an angle \(b\) with the positive x-axis (polar axis). It is graphed in polar coordinates by drawing a line from the origin, making an angle \(b\) with the positive x-axis.