Chapter 8: Problem 27
Convert the polar equation to rectangular form and sketch its graph. $$ r=3 $$
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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.) \(\frac{\text { Conic }}{\text { Ellipse }} \quad \frac{\text { Eccentricity }}{e=\frac{1}{2}} \quad \frac{\text { Directrix }}{x=1}\)
In Exercises 57 and \(58,\) let \(r_{0}\) represent the distance from the focus to the nearest vertex, and let \(r_{1}\) represent the distance from the focus to the farthest vertex. Show that the eccentricity of a hyperbola can be written as \(e=\frac{r_{1}+r_{0}}{r_{1}-r_{0}} .\) Then show that \(\frac{r_{1}}{r_{0}}=\frac{e+1}{e-1}\).
Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 \cos \theta, \quad y=6 \sin \theta $$
In Exercises 47 and 48, use the integration capabilities of a graphing utility to approximate to two decimal places the area of the surface formed by revolving the curve about the polar axis. $$ r=4 \cos 2 \theta, \quad 0 \leq \theta \leq \frac{\pi}{4} $$
Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Circle: } x=h+r \cos \theta, \quad y=k+r \sin \theta $$
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