Exam 12: Parametric Equations and Polar Coordinates

Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. -An ellipse with vertices $( \pm 6,0 )$ and foci at $( \pm 5,0 )$

Provide an appropriate response. -Graph the equation $r = 1 + a \cos \theta$ for several values of a in order to get an idea of what these curves look like. (Try both $| \mathrm { a } | < 1$ and $| \mathrm { a } | > 1$ .) Show that the graph will have an inner loop for every a such that $| \mathrm { a } | > 1$ and find the values of $\theta$ that correspond to the inner loop.

The polar equation of a circle is given. Give polar coordinates for the center of the circle and identify its radius. - $r = - 4 \cos \theta$

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

Find the eccentricity of the ellipse. - $4 x ^ { 2 } + y ^ { 2 } = 36$

Provide an appropriate response. -The area of the surface formed by revolving the graph of $r = f ( \theta )$ from $\theta = \alpha$ to $\theta = \beta$ about the line $\theta = 0$ (the polar axis) is $\mathrm { SA } = \int _ { \alpha } ^ { \beta } 2 \pi \mathrm { r } \sin \theta \sqrt { \mathrm { r } ^ { 2 } + \left( \frac { \mathrm { dr } } { \mathrm { d } \theta } \right) ^ { 2 } } \mathrm {~d} \theta$ . The area of the surface formed by revolving the graph of $\mathrm { r } = \mathrm { f } ( \theta )$ from $\theta = \alpha$ to $\theta = \beta$ about the line $\theta = \frac { \pi } { 2 }$ is $\mathrm { SA } = \int _ { \alpha } ^ { \beta } 2 \pi \mathrm { r } \cos \theta \sqrt { \mathrm { r } ^ { 2 } + \left( \frac { \mathrm { dr } } { \mathrm { d } \theta } \right) ^ { 2 } } \mathrm {~d} \theta$ . Use the idea underlying these two integral formulas to find the surface area of the torus formed by revolving the circle $r =$ a about the line $r = b \sec \theta$ where $0 < a < b$ .

Graph the polar equation. - $r=-\frac{3}{2}+\cos \theta$

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $x = \sin t , y = 6 \sin t , t = \frac { \pi } { 3 }$

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. - $e = \frac { 1 } { 3 } , y = 5$

Find the eccentricity of the ellipse. - $x ^ { 2 } + 2 y ^ { 2 } = 6$

Replace the Cartesian equation with an equivalent polar equation. -xy = 1

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

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - $r = \frac { 8 } { 2 - \cos \theta }$

Solve the problem. -The hyperbola $\frac { y ^ { 2 } } { 144 } - \frac { x ^ { 2 } } { 256 } = 1$ is shifted horizontally and vertically to obtain the hyperbola $\frac { ( y - 2 ) ^ { 2 } } { 144 } - \frac { ( x - 3 ) ^ { 2 } } { 256 } = 1$ Find the asymptotes of the new hyperbola.

Find the slope of the polar curve at the indicated point. - $r = 6 ( 1 + \cos \theta ) , \theta = \frac { \pi } { 4 }$

Find the area of the specified region. -Shared by the cardioids $r = 4 ( 1 + \sin \theta )$ and $r = 4 ( 1 - \sin \theta )$

Determine if the given polar coordinates represent the same point. - $( 1 , \pi / 6 ) , ( - 1,7 \pi / 6 )$

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. - $6 x ^ { 2 } - y ^ { 2 } - 72 x + 12 y - 114 = 0$

Graph the polar equation. - $r ^ { 2 } = 73 \sin 4 \theta$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $36 x ^ { 2 } - y ^ { 2 } = 36$