Problem 81
Question
For constants \(a\) and \(b\), describe the graphs of the equations \(r=a\) and \(\theta=b\) in polar coordinates.
Step-by-Step Solution
Verified 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.
1Step 1: 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.
2Step 2: 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.
Other exercises in this chapter
Problem 78
Use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Conchoid $$ r=2+\csc \theta $$ $$ y=1 $$
View solution Problem 81
Use the parametric equations $$ x=a(\theta-\sin \theta) $$ and \(\quad y=a(1-\cos \theta), a>0\) to answer the following. (a) Find \(d y / d x\) and \(d^{2} y /
View solution Problem 82
Use the parametric equations \(x=t^{2} \sqrt{3} \quad\) and \(\quad y=3 t-\frac{1}{3} t^{3}\) to answer the following. (a) Use a graphing utility to graph the c
View solution Problem 82
How are the slopes of tangent lines determined in polar coordinates? What are tangent lines at the pole and how are they determined?
View solution