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Chapter 8: Problem 27
Convert the polar equation to rectangular form and sketch its graph. $$ r=3 $$
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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 \(\frac{r_{1}}{r_{0}}=\frac{e+1}{e-1}\).
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 \(7-16,\) 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. \(r(3-2 \cos \theta)=6\)
Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t, \quad y=|t-2| $$
Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=e^{t}, \quad y=e^{3 t}+1 $$
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