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Chapter 8: Problem 77

Use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Conchoid $$ r=2-\sec \theta $$ $$ x=-1 $$

Short Answer

Expert verified
The Cartesian equation, obtained from the polar equation, can be graphed using a graphing tool. The vertical line \(x=-1\) identified in the graph respects the definition of an asymptote. Thus, \(x=-1\) is indeed an asymptote to the graph of the given polar equation \(r=2-\sec \theta\).

Step by step solution

01

Convert the Polar Equation to Cartesian Form

Given the polar equation \(r = 2 - \sec \theta\). Use the relationship between polar and Cartesian coordinates \(r = \sqrt{x^2+y^2}\) and \(\cos \theta = x/r\). Substitute these into the provided equation.
02

Simplify The Equation

After substitution, simplify the equation to obtain the Cartesian equation which will be helpful in graphing.
03

Graph the Equation and Identify the Asymptote

Utilize a graphing tool to graph the equation obtained above. An asymptote is a line that the curve gets closer to as it extends into infinity. Identify \(x=-1\) as this line in the graph.

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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. $$ x=e^{-t}, \quad y=e^{2 t}-1 $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=|t-1|, \quad y=t+2 $$

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. \(r=\frac{-6}{3+7 \sin \theta}\)

The curve represented by the equation \(r=a \theta,\) where \(a\) is a constant, is called the spiral of Archimedes. (a) Use a graphing utility to graph \(r=\theta,\) where \(\theta \geq 0\). What happens to the graph of \(r=a \theta\) as \(a\) increases? What happens if \(\theta \leq 0 ?\) (b) Determine the points on the spiral \(r=a \theta(a>0, \theta \geq 0)\) where the curve crosses the polar axis. (c) Find the length of \(r=\theta\) over the interval \(0 \leq \theta \leq 2 \pi\). (d) Find the area under the curve \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\).

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Ellipse: } x=h+a \cos \theta, \quad y=k+b \sin \theta $$
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