Chapter 10

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(3,-\frac{7 \pi}{6}\right)$$

Find the standard form of the equation of the circle with the given characteristics. Endpoints of a diameter: (1,-7) and (9,-5)

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (±3,0)\(;\) foci: (±6,0)

(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. $$x=t, y=-4 t$$

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(-4, \frac{5 \pi}{6}\right)$$

Identify the center and radius of the circle. $$x^{2}+y^{2}=36$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (2,0),(6,0)\(;\) foci: (0,0),(8,0)

(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. $$x=t, y=\frac{1}{2} t$$

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(-5,-\frac{11 \pi}{6}\right)$$

Identify the center and radius of the circle. $$x^{2}+y^{2}=121$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (2,3),(2,-3)\(;\) foci: (2,5),(2,-5)

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{3}{1-\cos \theta}$$

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r=3$$

(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. $$x=3 t-3, y=2 t+1$$

Identify the center and radius of the circle. $$(x-5)^{2}+y^{2}=9$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (4,1),(4,9)\(;\) foci: (4,0),(4,10)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (±3,0)\(;\) foci: (±2,0)

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{2}{4+\sin \theta}$$

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r=5+4 \cos \theta$$

(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. $$x=3-2 t, y=2+3 t$$

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(-5 \sqrt{2}, \frac{2 \pi}{3}\right)$$

Identify the center and radius of the circle. $$x^{2}+(y+8)^{2}=25$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (-2,1),(2,1)\(;\) foci: (-3,1),(3,1)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,±8)\(;\) foci: (0,±4)

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{4}{1-5 \cos \theta}$$

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r=\frac{2}{1-\cos \theta}$$

(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. $$x=\frac{1}{4} t, y=t^{2}$$

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$(0,-\pi / 2)$$

Identify the center and radius of the circle. $$(x+1)^{2}+(y+6)^{2}=19$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (2,3),(2,-3)\(;\) passes through the point (0,5)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (0,±5)\(;\) major axis of length 14

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{7}{1+\sin \theta}$$

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r=\frac{2}{1+\sin \theta}$$

(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. $$x=t, y=t^{3}$$

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$(0, \pi / 6)$$

Identify the center and radius of the circle. $$(x+7)^{2}+(y-3)^{2}=32$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (-2,1),(2,1)\(;\) passes through the point (5,4)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (±2,0)\(;\) major axis of length 10

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

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r=6 \sin \theta$$

(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. $$x=t+2, y=t^{2}$$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$(3, \pi)$$

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

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (0,4),(0,0) passes through the point \((\sqrt{5},-1)\)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,±4) passes through the point (3,1)

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{9}{3-2 \cos \theta}$$

Test for symmetry with respect to the line \(\theta=\pi / 2,\) the polar axis, and the pole. $$r=4 \csc \theta \cos \theta$$

(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. $$x=\sqrt{t}, y=1-t$$

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$(2,0)$$

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