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Chapter 8: Problem 81

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

Short Answer

Expert verified
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

01

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.
02

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.

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Most popular questions from this chapter

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Witch of Agnesi: } x=2 \cot \theta, \quad y=2 \sin ^{2} \theta $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{5}{5+3 \sin \theta}\)

In Exercises 49 and 50 , use the integration capabilities of a graphing utility to approximate to two decimal places the area of the region bounded by the graph of the polar equation. \(r=\frac{2}{3-2 \sin \theta}\)

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{3}{-4+2 \sin \theta}\)

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=e^{a \theta} & 0 \leq \theta \leq \frac{\pi}{2} & \theta=\frac{\pi}{2} \end{array} $$
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