Exam 12: Parametric Equations and Polar Coordinates

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 144 } = 1$ is shifted horizontally and vertically to obtain the ellipse $\frac { ( x - 3 ) ^ { 2 } } { 400 } + \frac { ( y + 3 ) ^ { 2 } } { 144 } = 1$ Find the foci of the new ellipse.

Solve the problem. -The parabola $y ^ { 2 } = - 10 x$ is shifted down 5 units and left 7 units. Find an equation for the new parabola and find the new vertex.

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Vertices at $( 0,4 )$ and $( 0 , - 4 )$ ; asymptotes $y = \frac { 2 } { 3 } x$ and $y = - \frac { 2 } { 3 } x$

Provide an appropriate response. -Find any horizontal or vertical asymptotes for the spiral $r = \frac { 1 } { \theta } , \theta > 0$ . (Suggestion: Express the curve in parametric form $( \mathrm { x } = \mathrm { r } \cos \theta , \mathrm { y } = \mathrm { r } \sin \theta )$ and investigate what happens to $\mathrm { x }$ and $\mathrm { y }$ as $\theta$ varies. $)$

Find the vertices and foci of the ellipse. - $49 x ^ { 2 } + 25 y ^ { 2 } = 1225$

Solve the problem. -The hyperbola $\frac { y ^ { 2 } } { 11 } - \frac { x ^ { 2 } } { 5 } = 1$ is shifted down 7 units and right 5 units. Find an equation for the new hyperbola and find the new vertices.

Find the slope of the polar curve at the indicated point. - $r = - 8 + 5 \cos \theta , \theta = \frac { \pi } { 2 }$

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - $r = \frac { 12 } { 4 + 4 \sin \theta }$

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

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$ -(4, -4)

Replace the Cartesian equation with an equivalent polar equation. -x = 11

Replace the Cartesian equation with an equivalent polar equation. - $( x - 18 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 324$

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 5 \sin 4 \theta , \theta = 0$

Describe the graph of the polar equation. - $r ^ { 2 } + 2 r ^ { 2 } \sin \theta \cos \theta = 144$

Solve the problem. -The hyperbola $\frac { x ^ { 2 } } { 100 } - \frac { y ^ { 2 } } { 64 } = 1$ is shifted horizontally and vertically to obtain the hyperbola $\frac { ( x + 5 ) ^ { 2 } } { 100 } - \frac { ( y + 2 ) ^ { 2 } } { 64 } = 1$ Find the center and vertices of the new hyperbola.

Find the focus and directrix of the parabola. - $y = 8 x ^ { 2 }$

Find a Cartesian equation for the line whose polar equation is given. - $r \cos \left( \theta - \frac { 5 \pi } { 6 } \right) = 5$

Find the slope of the polar curve at the indicated point. - $r = \frac { 5 } { \theta } , \theta = 4 \pi$

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Foci at $( \sqrt { 17 } , 0 ) , ( - \sqrt { 17 } , 0 )$ ; asymptotes $y = 4 x , y = - 4 x$

Find the eccentricity of the ellipse. -Find the eccentricity of an ellipse centered at the origin having a focus of $( - \sqrt { 11 } , 0 )$ and corresponding directrix $x = - \frac { 36 } { \sqrt { 11 } }$