91Ó°ÊÓ

Find the areas of the surfaces generated by revolving the curves about the indicated axes. \(r=\sqrt{\cos 2 \theta}, \quad 0 \leq \theta \leq \pi / 4, \quad y\) -axis

In Exercises \(23-30\) , find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$ 8 y^{2}-2 x^{2}=16 $$

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

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r \sin \theta=r \cos \theta $$

Find the areas of the surfaces generated by revolving the curves about the indicated axes. \(r=\sqrt{2} e^{\theta / 2}, \quad 0 \leq \theta \leq \pi / 2, \quad x\) -axis

In Exercises \(23-30\) , find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$ 64 x^{2}-36 y^{2}=2304 $$

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

Show that \((1 / 2,3 \pi / 2)\) lies on the curve \(r=-\sin (\theta / 3)\)

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. $$ 8 x^{2}-2 y^{2}=16 $$