Exam 12: Parametric Equations and Polar Coordinates

Find the standard-form equation for an ellipse which satisfies the given conditions. -An ellipse centered at the origin having vertex at $( 0 , - 9 )$ and eccentricity equal to $\frac { 1 } { 3 }$

Graph. - $x ^ { 2 } - y ^ { 2 } = 8$

Find the focus and directrix of the parabola. - $y ^ { 2 } = - 20 x$

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$ - $y = - 4$

Solve the problem. -The hyperbola $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 9 } = 1$ is shifted horizontally and vertically to obtain the hyperbola $\frac { ( x + 2 ) ^ { 2 } } { 64 } - \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$ . Graph the new hyperbola.

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

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

Replace the polar equation with an equivalent Cartesian equation. - $r ^ { 2 } = 44 r \cos \theta$

Find the length of the curve. -The spiral $r = 6 \theta , 0 \leq \theta \leq \pi$

Graph the polar equation. - $r = 4 \sin 2 \theta$

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. - $x=4 t^{2}, y=2 t, x \leq t \leq \infty$

Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. -An ellipse with vertices $( 0 , \pm 9 )$ and foci at $( 0 , \pm 2 \sqrt { 3 } )$ .

Determine if the given polar coordinates represent the same point. - $( 4 , \pi / 2 ) , ( - 4,3 \pi / 2 )$

Find the foci of the hyperbola. - $x ^ { 2 } - y ^ { 2 } = 50$

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$ - $x - \sqrt { 3 } y = 4$

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$ - $\left( 0 , \frac { 1 } { 5 } \right)$

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

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$ - $\sqrt { 2 } x - \sqrt { 2 } y = 10$

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. - $x ^ { 2 } + y ^ { 2 } - 4 x = 0$