Exam 12: Parametric Equations and Polar Coordinates

Solve the problem. -The parabola $x ^ { 2 } = - 32 y$ is shifted 2 units up and 6 units to the left. Graph the new parabola.

Find the standard-form equation for an ellipse which satisfies the given conditions. -An ellipse centered at the origin having focus $( \sqrt { 77 } , 0 )$ and directrix $x = \frac { 81 } { \sqrt { 77 } }$

Find the vertices and foci of the ellipse. - $\frac { x ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 625 } = 1$

If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. - $\mathrm { x } ^ { 2 } + 6 \mathrm { x } - 4 \mathrm { y } + 17 = 0$

Find the length of the curve. -The line segment $r = 9 \sec \theta , 0 \leq \theta \leq \frac { \pi } { 4 }$

Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. -An ellipse with length of major axis 18 and $y$ -intercepts $( 0 , \pm 5 )$

Provide an appropriate response. - $\text { Determine whether or not the point } ( 1 , \pi ) \text { lies on the curve } r ^ { 2 } \cos \theta = 1 \text {. Explain. }$

Find the area of the surface generated by revolving the curves about the indicated axis. - $x = \sin t , y = 2 + \cos t , 0 \leq t \leq 2 \pi ; x$ -axis

Choose the equation that matches the graph. -

Find the area. -Find the area under one arch of the cycloid x = 2(t - sin t), y = 2(1 - cos t).

Choose the equation that matches the graph. -

Find the Cartesian coordinates of the given point. -( , 0)

Provide an appropriate response. - $\text { Find the left most point on the cardioid } \mathrm { r } = 1 + \sin \theta \text {. }$

Find a Cartesian equation for the line whose polar equation is given. - $r \cos \left( \theta - \frac { 3 \pi } { 4 } \right) = 5 \sqrt { 2 }$

Determine if the given polar coordinates represent the same point. - $( 5 , - 4 \pi / 3 ) , ( - 5,5 \pi / 3 )$

Find the directrices of the ellipse. - $144 x ^ { 2 } + 4 y ^ { 2 } = 576$

Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t. - $x ( t + 1 ) - 4 t \sqrt { x } = 36,2 y + 4 y ^ { 3 / 2 } = t ^ { 3 } + t , t = 0$

Solve the problem. -The hyperbola $\frac { y ^ { 2 } } { 400 } - \frac { x ^ { 2 } } { 225 } = 1$ is shifted horizontally and vertically to obtain the hyperbola $\frac { ( y - 2 ) ^ { 2 } } { 400 } - \frac { ( x + 4 ) ^ { 2 } } { 225 } = 1$ Find the foci of the new hyperbola.

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. - $e = \frac { 1 } { 5 } , y = - 10$

Find a polar equation for the circle. - $x ^ { 2 } + ( y + 3 ) ^ { 2 } = 9$