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 focus $( \sqrt { 35 } , 0 )$ and directrix $x = \frac { 36 } { \sqrt { 35 } }$

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. - $16 x ^ { 2 } + 128 x + 25 y ^ { 2 } + 250 y + 481 = 0$

Graph the parabola. - $y ^ { 2 } = - 2 x$

Find the area. - $\text { Find the area of the region between the curve } x = e ^ { 5 t } , y = \frac { 1 } { 5 } e ^ { - 4 t } \text { and the } x \text {-axis, } 0 \leq t \leq \ln 9 \text {. }$

Describe the graph of the polar equation. - $r \cos \theta = 10$

Find the area of the surface generated by revolving the curves about the indicated axis. - $x = t + \sqrt { 30 } , y = \frac { t ^ { 2 } } { 2 } + \sqrt { 30 } t , - \sqrt { 30 } \leq t \leq \sqrt { 30 } ; y$ -axis

Solve the problem. -The ellipse $\frac { x ^ { 2 } } { 400 } + \frac { y ^ { 2 } } { 256 } = 1$ is shifted horizontally and vertically to obtain the ellipse $\frac { ( x + 4 ) ^ { 2 } } { 400 } + \frac { ( y + 5 ) ^ { 2 } } { 256 } = 1$ Find the center and vertices of the new ellipse.

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $x = t + \cos t , y = 2 - \sin t , t = \frac { \pi } { 6 }$

Find the foci of the ellipse. - $49 x ^ { 2 } + 64 y ^ { 2 } = 3136$

Find the area of the specified region. -Inside the limacon $r = 7 + 6 \sin \theta$

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $x = 4 \sin t , y = 4 \cos t , t = \frac { 3 \pi } { 4 }$

Replace the polar equation with an equivalent Cartesian equation. - $8 r \cos \theta + 9 r \sin \theta = 1$

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

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

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. - $x ^ { 2 } + y ^ { 2 } - 14 x + 18 y = - 114$

Replace the polar equation with an equivalent Cartesian equation. - $r ^ { 2 } + 2 r ^ { 2 } \sin \theta \cos \theta = 625$

Find the Cartesian coordinates of the given point. - $( 9 , \pi )$

Find the area of the specified region. -Inside the lemniscate $r ^ { 2 } = 4 \sin 2 \theta$ and outside the circle $r = \sqrt { 2 }$

Solve the problem. -Find the coordinates of the centroid of the area bounded by $y = x ^ { 2 }$ and $y = 3$

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. - $x \cos t + x = 2 t , y = t \sin t + t , t = \frac { \pi } { 2 }$