Chapter 11: Problem 11
What are the equations of the asymptotes of a standard hyperbola with vertices on the \(x\) -axis?
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A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and \(3,\) respectively, and that Earth completes one orbit in one year while Mars takes two years. The position of Mars as seen from Earth is given by the parametric equations $$x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t$$ a. Graph the parametric equations, for \(0 \leq t \leq 2\) b. Letting \(r=(3-4 \cos \pi t),\) explain why the path of Mars as seen from Earth is a limaçon.
Points at which the graphs of \(r=f(\theta)\) and \(r=g(\theta)\) intersect must be determined carefully. Solving \(f(\theta)=g(\theta)\) identifies some-but perhaps not all-intersection points. The reason is that the curves may pass through the same point for different values of \(\theta .\) Use analytical methods and a graphing utility to find all the intersection points of the following curves. $$r=1-\sin \theta \text { and } r=1+\cos \theta$$
Sketch the graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$12 x=5 y^{2}$$
Find a polar equation for each conic section. Assume one focus is at the origin.
Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{6}{3+2 \sin \theta}$$
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