Exam 12: Parametric Equations and Polar Coordinates

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $0 \leq \theta \leq \pi , r \leq 3$

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Foci at $( - 20,0 ) , ( 20,0 )$ ; asymptotes: $y = \frac { 3 } { 4 } x , y = - \frac { 3 } { 4 } x$

Replace the polar equation with an equivalent Cartesian equation. - $\mathrm { r } ^ { 2 } = 50 \mathrm { r } \cos \theta - 6 \mathrm { r } \sin \theta - 9$

Find the length of the curve. - $x = \frac { 2 } { 3 } \left( t ^ { 2 } + 2 \right) ^ { 3 / 2 } , y = 2 t , 0 \leq t \leq 1$

Graph. - $49 x^{2}+4 y^{2}=196$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x = 4 ( t - \sin t ) , y = 4 ( 1 - \cos t ) , 0 \leq t \leq 4 \pi$

Graph. - $16 x^{2}+36 y^{2}=576$

Plot the point whose polar coordinates are given. - $(-2, \pi / 2)$

Replace the Cartesian equation with an equivalent polar equation. - $x ^ { 2 } + y ^ { 2 } = 100$

Determine the symmetries of the curve. - $r = - 2 \sin \theta - 2 \cos \theta$

Graph the polar equation. - $r = 3 ( 2 + 2 \sin \theta )$

Find the eccentricity of the ellipse. -Find the eccentricity of an ellipse centered at the origin having a focus of $( 0 , \sqrt { 35 } )$ and corresponding directrix $y$ $= \frac { 36 } { \sqrt { 35 } }$

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Asymptotes $y = \frac { 6 } { 7 } x , y = - \frac { 6 } { 7 } x$ ; one vertex is $( 7,0 )$

Find the eccentricity of the hyperbola. - $x ^ { 2 } - 2 y ^ { 2 } = 6$

Find the area of the specified region. -Inside the circle $r = 8 \cos \theta + 9 \sin \theta$

Find the length of the curve. -The circle $r = 6 \sin \theta$

Replace the polar equation with an equivalent Cartesian equation. - $\mathrm { r } = 7 \cot \theta \csc \theta$

Provide an appropriate response. -Consider the curves corresponding to polar equations of the form $r = \frac { 1 - a \cos \theta } { 1 + a \cos \theta }$ where $a > 0$ . Explain how the graph changes as a changes. Identify any values of a for which the basic shape of the curve changes.

Provide an appropriate response. -How are the graphs of $r = 1 + \cos ( \theta - \pi / 6 )$ and $r = 1 + \cos ( \theta - \pi / 4 )$ related to the graph of $r = 1 + \cos \theta$ ? In general, how is the graph of $\mathrm { r } = \mathrm { f } ( \theta - \alpha )$ related to the graph of $\mathrm { r } = \mathrm { f } ( \theta )$ ?

Find the eccentricity of the ellipse. - $25 x ^ { 2 } + 64 y ^ { 2 } = 1600$