Problem 107
For the following exercises, find the Cartesian equation describing the given shapes. A hyperbola with vertices at \((3,-2)\) and \((-5,-2)\) and foci at \((-2,-6)\) and \((-2,4)\)
Problem 108
For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. $$ r=\frac{6}{1+3 \cos (\theta)} $$
Problem 109
For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. $$ r=\frac{4}{3-2 \cos \theta} $$
Problem 110
For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. $$ r=\frac{7}{5-5 \cos \theta} $$
Problem 111
Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is \(39.26 \mathrm{AU}\) and minor axis is \(38.07 \mathrm{AU}\). What is the eccentricity?
Problem 112
The C/1980 E1 comet was observed in 1980 . Given an eccentricity of \(1.057\) and a perihelion (point of closest approach to the Sun) of \(3.364 \mathrm{AU}\), find the Cartesian equations describing the comet's trajectory. Are we guaranteed to see this comet again? (Hint: Consider the Sun at point \((0,0)\).)
