Exam 12: Parametric Equations and Polar Coordinates

Solve the problem. -The hyperbola $\frac { x ^ { 2 } } { 400 } - \frac { y ^ { 2 } } { 225 } = 1$ is shifted horizontally and vertically to obtain the hyperbola $\frac { ( x + 5 ) ^ { 2 } } { 400 } - \frac { ( y - 3 ) ^ { 2 } } { 225 } = 1$ Find the asymptotes of the new hyperbola.

Determine the symmetries of the curve. - $r ^ { 2 } = 6 \sin 3 \theta$

Find the area of the specified region. -Inside the six-leaved rose $r ^ { 2 } = 5 \cos 3 \theta$

Describe the graph of the polar equation. - $\mathrm { r } = - 11 \csc \theta$

Find the length of the curve. - $x = 7 \sin ^ { 3 } t , y = 7 \cos ^ { 3 } t , 0 \leq t \leq \pi$

Find the slope of the polar curve at the indicated point. - $r = 6 \cos 3 \theta , \theta = \frac { \pi } { 3 }$

Find the length of the curve. -The curve $r = a \cos ^ { 2 } \left( \frac { \theta } { 2 } \right) , 0 \leq \theta \leq \pi , a > 0$

Find the foci of the ellipse. - $9 x ^ { 2 } + 25 y ^ { 2 } = 225$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x = 5 \cos 2 t + 4 \cos 6 t , y = 5 \sin 2 t - 4 \sin 6 t , 0 \leq t \leq \pi$

Find the area of the specified region. -Inside the circle $r = - 6 \cos \theta$ and outside the circle $r = 3$

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$ - $\left( \frac { 1 } { 3 } , \frac { - \sqrt { 3 } } { 3 } \right)$

Find a polar equation for the circle. - $( x + 9 ) ^ { 2 } + y ^ { 2 } = 81$

Solve the problem. -The hyperbola $\frac { x ^ { 2 } } { 5 } - \frac { y ^ { 2 } } { 16 } = 1$ is shifted down 8 units and right 2 units. Find an equation for the new hyperbola and find the new center.

Describe the graph of the polar equation. - $r ^ { 2 } = 46 r \cos \theta$

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - $r=\frac{15}{3-3 \sin \theta}$

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$ is shifted up 6 units and left 8 units. Find an equation for the new ellipse and find the new center.

Find all the polar coordinates of the point. - $( - 6 , \pi / 3 )$

The polar equation of a circle is given. Give polar coordinates for the center of the circle and identify its radius. - $\mathrm { r } = 10 \sin \theta$

Replace the Cartesian equation with an equivalent polar equation. - $x ^ { 2 } - y ^ { 2 } = 4$

Find the slope of the polar curve at the indicated point. - $r = 5 \cos \theta + 8 \sin \theta , \theta = 0$