91Ó°ÊÓ

Which of the following has the same graph as \(r=\cos 2 \theta ?\) a. \(r=-\sin (2 \theta+\pi / 2) \quad\) b. \(\quad r=-\cos (\theta / 2)\) Confirm your answer with algebra.

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=1, \quad y=2$$

Exercises \(27-34\) give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch. $$ y^{2}-x^{2}=4 $$

Find the lengths of the curves. $$ \begin{array}{l}{x=\ln (\sec t+\tan t)-\sin t} \\ {y=\cos t, \quad 0 \leq t \leq \pi / 3}\end{array} $$

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \cos \theta=0$$

Find the areas of the surfaces generated by revolving the curves about the indicated axes. $$ x=\cos t, \quad y=2+\sin t, \quad 0 \leq t \leq 2 \pi ; \quad x-\text {axis} $$

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=5, \quad y=-6$$

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r=4 \csc \theta$$

Exercises \(27-34\) give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch. $$ 8 x^{2}-2 y^{2}=16 $$

A rose within a rose Graph the equation \(r=1-2 \sin 3 \theta.\)