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Chapter 8: Problem 62

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=3(1-\cos \theta) $$

Short Answer

Expert verified
The tangents to the graph of \(r = 3(1 - \cos \theta)\) at the pole are the positive and negative x-axes, i.e., the lines at \(\theta = 0\) and \(\theta = \pi\).

Step by step solution

01

Sketch the graph

Start by plotting values for \(\theta\) from 0 to \(2 \pi\) in increments, say every \( \pi /4\), and calculating the corresponding values of r using the given equation \(r = 3(1 - \cos \theta)\). A collection of points is obtained, which can be plotted and then connected smoothly to reveal the graph. Notice that the graph is discontinuous at \(\theta = \pi\).
02

Find the pole

Look for the point on the graph where r=0. This is the pole. According to the equation, \(r = 3(1 - \cos \theta)\), the pole is located at \(\theta = 0\).
03

Determine the tangents at the pole

The tangent lines at the pole are lines that barely touch the graph at that point, without crossing it. Usually, this involves determining the slope of the graph at that point, however in the case of a pole the graph is discontinuous at the pole. Because the tangent cannot cross the graph, the tangents to the graph at \(\theta = 0\) only occur along the positive and negative x-axes, i.e., the lines \(\theta = 0\) and \(\theta = \pi\).

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Most popular questions from this chapter

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t^{3}, \quad y=\frac{t^{2}}{2} $$

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \begin{aligned} &\text { Line through }\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)\\\ &x=x_{1}+t\left(x_{2}-x_{1}\right), \quad y=y_{1}+t\left(y_{2}-y_{1}\right) \end{aligned} $$

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{-1}{1-\cos \theta}\)

Writing Consider the polar equation \(r=\frac{4}{1+e \sin \theta} .\) (a) Use a graphing utility to graph the equation for \(e=0.1\), \(e=0.25, e=0.5, e=0.75,\) and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) (b) Use a graphing utility to graph the equation for \(e=1\). Identify the conic. (c) Use a graphing utility to graph the equation for \(e=1.1\), \(e=1.5,\) and \(e=2 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{+}\) and \(e \rightarrow \infty\).

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Hyperbola }} \quad \frac{\text { Eccentricity }}{e=\frac{3}{2}} \quad \frac{\text { Directrix }}{x=-1}\)
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