Chapter 10

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (1,2),(1,-2) passes through the point \((0, \sqrt{5})\)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (±8,0) passes through the point (5,-3)

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{6}{2+\sin \theta}$$

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r^{2}=16 \sin 2 \theta$$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(1,-\frac{3 \pi}{4}\right)$$

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=2 t, y=|t-2|$$

Write the equation of the circle in standard form. Then identify its center and radius. $$\frac{4}{3} x^{2}+\frac{4}{3} y^{2}=1$$

Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. Use graphing utility to verify your graph. \(x^{2}-y^{2}=1\)

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{5}{-1+\cos \theta}$$

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r^{2}=36 \sin 2 \theta$$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(-16, \frac{5 \pi}{2}\right)$$

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=|t-1|, y=t+2$$

Write the equation of the circle in standard form. Then identify its center and radius. $$\frac{9}{2} x^{2}+\frac{9}{2} y^{2}=1$$

Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. Use graphing utility to verify your graph \(y^{2}-x^{2}=1\)

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{3}{-4-8 \cos \theta}$$

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=5$$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(0,-\frac{7 \pi}{6}\right)$$

Write the equation of the circle in standard form. Then identify its center and radius. $$x^{2}+y^{2}-2 x+6 y+9=0$$

Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. Use graphing utility to verify your graph \(\frac{y^{2}}{1}-\frac{x^{2}}{16}=1\)

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (0,2),(8,2)\(;\) minor axis of length 2

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{10}{3+9 \sin \theta}$$

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$\theta=-5 \pi / 3$$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(0, \frac{5 \pi}{4}\right)$$

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=\cos \theta, y=4 \sin \theta$$

Write the equation of the circle in standard form. Then identify its center and radius. $$x^{2}+y^{2}-10 x-6 y+25=0$$

Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. Use graphing utility to verify your graph \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\)

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (3,1),(3,9)\(;\) minor axis of length 6

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{5}{1-\sin \theta}$$

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=3 \sin \theta$$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(-5, \frac{3 \pi}{2}\right)$$

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=e^{-t}, y=e^{3 t}$$

Write the equation of the circle in standard form. Then identify its center and radius. $$4 x^{2}+4 y^{2}+12 x-24 y+41=0$$

Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. Use graphing utility to verify your graph \(\frac{y^{2}}{16}-\frac{x^{2}}{4}=1\)

Find the standard form of the equation of the ellipse with the given characteristics. Foci: (0,0),(0,8)\(;\) major axis of length 36

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{-1}{2+4 \sin \theta}$$

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=2 \cos \theta$$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(-3,-\frac{\pi}{6}\right)$$

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=e^{2 t}, y=e^{t}$$

Write the equation of the circle in standard form. Then identify its center and radius. $$9 x^{2}+9 y^{2}+54 x-36 y+17=0$$

Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. Use graphing utility to verify your graph \(\frac{x^{2}}{25}-\frac{y^{2}}{36}=1\)

Find the standard form of the equation of the ellipse with the given characteristics. Foci: (0,0),(4,0)\(;\) major axis of length 6

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{14}{14+17 \sin \theta}$$

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=3(1-\cos \theta)$$

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(2,2 \pi / 9)$$

Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. Use graphing utility to verify your graph \(\frac{(x-3)^{2}}{9}-\frac{(y-1)^{2}}{1}=1\)

Find the standard form of the equation of the ellipse with the given characteristics. $$\text { Center: }(3,2) ; a=3 c ; \text { foci: }(1,2),(5,2)$$

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{12}{2-\cos \theta}$$

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=4(1+\sin \theta)$$

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(4,11 \pi / 9)$$

(a) sketch the curve represented by the parametric equations (indicate the orientation of the curve). Use a graphing utility to confirm your result. (b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. $$x=\ln 2 t, y=2 t^{2}$$