Exam 9: Topics in Analytic Geometry

Sketch the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $r = 2 - 3 \cos \theta$ Use either grid below for your graph, whichever is more convenient.

Find the standard form of the equation of the hyperbola with the given characteristics. foci: $( \pm 3,0 ) \quad$ asymptotes: $y = \pm 4 x$

Find the center and foci of the following ellipse. $4 x ^ { 2 } + 8 y ^ { 2 } - 16 x + 32 y + 32 = 0$

Match the graph with its equation.

Identify the center and radius of the circle below. $x ^ { 2 } + y ^ { 2 } + 10 x + 10 y - 5 = 0$

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = - 6 + \cos ( 5 \theta )$

Rotate the axes to eliminate the xy-term in the following equation and then write the equation in standard form. $3 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 0$

Find an interval for $\theta$ for which the graph is traced only once. $r = 2 \sin \left( \frac { 3 \theta } { 2 } \right)$

Find the standard form of the equation of the hyperbola below. $8 x ^ { 2 } - y ^ { 2 } - 80 x - 16 y + 96 = 0$

Rotate the axes to eliminate the $x y$ -term in the following equation. $- 30 x ^ { 2 } + 20 \sqrt { 3 } x y - 10 y ^ { 2 } - 20 x - 20 \sqrt { 3 } y = 0$

Find the rectangular graph of the following polar equation. $\theta=\frac{5 \pi}{4}$

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Vertices Hyperbola , , -2,

Find the graph of the following polar equation. $r = 6 \sin ( 2 \theta )$

Rotate the axes to eliminate the $x y$ -term in the following equation. $18 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 0$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $64 x ^ { 2 } + 64 y ^ { 2 } + 192 x + 24 y - 327 = 0$

Identify the center and radius of the circle below. $x ^ { 2 } + y ^ { 2 } + 6 x + 6 y - 5 = 0$

Which answer is a polar form of the given rectangular equation? $x ^ { 2 } + y ^ { 2 } = 25$

Find the graph of the following polar equation. $r = 2 \sin ( 2 \theta )$

Rotate the axes to eliminate the $x y$ -term in the equation below and then write the equation in standard form. $x y + 1 = 0$

Use a graphing utility to graph the rotated conic. $r = \frac { 1 } { 1 + 2 \cos ( \theta + \pi / 4 ) }$ Use either grid below for your graph, whichever is more convenient.