Exam 12: Parametric Equations and Polar Coordinates

Solve the problem. -The parabola $y ^ { 2 } = - 20 x$ is shifted 7 units down and 2 units to the right. Find the focus and directrix of the new parabola.

Find the area of the specified region. -Inside the cardioid $r = \alpha ( 1 + \sin \theta ) , \alpha > 0$

Choose the equation that matches the graph. -

Find a Cartesian equation for the line whose polar equation is given. - $r \cos \left( \theta - \frac { \pi } { 6 } \right) = 2$

The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. - $e = \frac { 1 } { 4 } , x = 2$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $\frac { y ^ { 2 } } { 400 } - \frac { x ^ { 2 } } { 225 } = 1$

Find the eccentricity of the hyperbola. - $9 x ^ { 2 } - 16 y ^ { 2 } = 144$

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Vertices at (0, 6) and (0, -6); foci at (0, 9) and (0, -9)

Sketch the region which is defined by the given conditions. - $\pi \leq \theta \leq \frac { 3 \pi } { 2 } , r = - 4$

Find the length of the curve. - $x = \cos 2 t , y = \sin 2 t , 0 \leq t \leq 2 \pi$

Solve the problem. -Find the volume of the solid generated by revolving the region enclosed by the ellipse $4 x ^ { 2 } + 25 y ^ { 2 } = 100$ about the $x$ -axis.

Find the length of the curve. - $x = \frac { 1 } { 3 } \left( t ^ { 3 } - 3 t \right) , y = t ^ { 2 } + 6,0 \leq t \leq 2$

Sketch the region which is defined by the given conditions. - $0 \leq \theta \leq \frac { \pi } { 4 } , 0 \leq \mathrm { r } \leq 4$

Find the focus and directrix of the parabola. - $y ^ { 2 } = 36 x$

$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$ - $x = \sqrt { t + 3 } , y = - t , t = 46$

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. - $x=2 \sin t, y=5 \cos t, 0 \leq t \leq 2 \pi$

Graph the polar equation. - $\mathrm { r } = 4 ( 1 - \sin \theta )$

Find the standard-form equation for a hyperbola which satisfies the given conditions. -A hyperbola centered at the origin having vertex at (0, -8) and eccentricity equal to 2

Find a Cartesian equation for the line whose polar equation is given. - $\mathrm { r } \cos \left( \theta + \frac { \pi } { 4 } \right) = 5 \sqrt { 2 }$

Provide an appropriate response. -Find the error in the following "proof" that the area inside the lemniscate $r ^ { 2 } = a ^ { 2 } \cos 2 \theta$ is 0 and then find the correct area. $\left. A = \int _ { 0 } ^ { 2 \pi } \frac { 1 } { 2 } r ^ { 2 } d \theta = \int _ { 0 } ^ { 2 \pi } \frac { 1 } { 2 } a ^ { 2 } \cos 2 \theta d \theta = \frac { 1 } { 4 } a ^ { 2 } \sin 2 \theta \right] _ { 0 } ^ { 2 \pi } = 0$