Chapter 8: Problem 39
Find two different sets of parametric equations for the rectangular equation. $$ y=3 x-2 $$
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Use the results of Exercises \(31-34\) to find a set of parametric equations for the line or conic. $$ \text { Ellipse: vertices: }(\pm 5,0) ; \text { foci: }(\pm 4,0) $$
True or False. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of the parametric equations \(x=t^{2}\) and \(y=t^{2}\) is the line \(y=x\).
Identify each conic. (a) \(r=\frac{5}{1-2 \cos \theta}\) (b) \(r=\frac{5}{10-\sin \theta}\) (c) \(r=\frac{5}{3-3 \cos \theta}\) (d) \(r=\frac{5}{1-3 \sin (\theta-\pi / 4)}\)
Explain why finding points of intersection of polar graphs may require further analysis beyond solving two equations simultaneously
Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? $$ \text { (a) } \begin{aligned} x &=2 \cos \theta \\ y &=2 \sin \theta \end{aligned} $$ $$ \begin{aligned} &\text { (b) } x=\sqrt{4 t^{2}-1} /|t|\\\ &y=1 / t \end{aligned} $$ $$ \text { (c) } \begin{aligned} x &=\sqrt{t} \\ y &=\sqrt{4-t} \end{aligned} $$ $$ \text { (d) } \begin{aligned} x &=-\sqrt{4-e^{2 t}} \\ y &=e^{t} \end{aligned} $$
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