(Solved): 1. Find the Cartesian coordinates of the following points given in polar coordinates. (a) (3 ...

1. Find the Cartesian coordinates of the following points given in polar coordinates. (a) $\left(-3,\frac{5\pi }{6}\right)$ (b) $\left(5,{\tan }^{-1}\left(\frac{4}{3}\right)\right)$ (c) $(-1,7\pi )$ (d) $\left(2\sqrt{3},\frac{2\pi }{3}\right)$ 2. Graph the sets of points whose polar coordinates satisfy the equations and inequalities in each of the following. (a) $\frac{\pi }{4}\le \theta \le \frac{3\pi }{4},0\le r\le 1$ (b) $-\frac{\pi }{4}\le \theta \le \frac{\pi }{4},-1\le r\le 1$ (c) $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2},1\le r\le 2$ (d) $0\le \theta \le \frac{\pi }{2},1\le |r|\le 2$ 3. In the following, you are given the eccentricities of conic sections with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for each conic section. (a) $e=\frac{1}{2},x=1$ (b) $e=\frac{1}{4},x=-2$ (c) $e=\frac{1}{5},x=-10$ (d) $e=\frac{1}{3},y=6$ 4. Sketch the regions defined by the following inequalities (a) $0\le r\le 2\cos \theta$ (b) $-3\cos \theta \le r\le 0$ 5. Graph the following conic sections (a) $r=\frac{8}{4+\cos \theta }$ (b) $r=\frac{8}{4+\sin \theta }$ (c) $r=\frac{1}{1-\sin \theta }$ (d) $r=\frac{4}{1+\cos \theta }$ (e) $r=\frac{1}{1+2\sin \theta }$ (f) $r=\frac{1}{1+2\cos \theta }$ 6. Find the Cartesian equations for the curves $r=4\sin \theta$ and $r=\sqrt{3}\sec \theta$. Sketch the curves together and label their points of intersection in both Cartesian and polar coordinates. 7. Find the Cartesian equations for the curves $r=8\cos \theta$ and $r=2\sec \theta$. Sketch the curves together and label their points of intersection in both Cartesian and polar coordinates. 8. Find a polar equation for the parabola with focus $(0,0)$ and directrix $r\cos \theta =4$. 9. Find a polar equation for the parabola with focus $(0,0)$ and directrix $r\cos \left(\theta -\frac{\pi }{2}\right)=2$.