Chapter 10

Sketch the circle. Identify its center and radius. $$y^{2}=81-x^{2}$$

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 \(\frac{(x-2)^{2}}{4}-\frac{(y+5)^{2}}{25}=1\)

Find the standard form of the equation of the ellipse with the given characteristics. $$\text { Center: }(0,4), a=5 c ; \text { vertices: }(0,-1),(0,9)$$

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{-3}{-4+2 \cos \theta}$$

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=4+5 \sin \theta$$

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(-4.5,1.3)$$

Use a graphing utility to graph the curve represented by the parametric equations. Use the graph and the Vertical Line Test to determine whether \(y\) is a function of \(x.\) $$\begin{array}{l} x=4+3 \cos \theta \\ y=-2+\sin \theta \end{array}$$

Sketch the circle. Identify its center and radius. $$x^{2}+8 x+y^{2}+2 y+8=0$$

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 \(\frac{(y+6)^{2}}{1}-\frac{(x-2)^{2}}{\frac{1}{16}}=1\)

Find the center, vertices, foci, and eccentricity of the ellipse, and sketch its graph. Use a graphing utility to verify your graph. $$\frac{x^{2}}{64}+\frac{y^{2}}{9}=1$$

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

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=3+6 \cos \theta$$

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(8.25,3.5)$$

Use a graphing utility to graph the curve represented by the parametric equations. Use the graph and the Vertical Line Test to determine whether \(y\) is a function of \(x.\) $$\begin{aligned} &x=4+3 \cos \theta\\\ &y=-2+2 \sin \theta \end{aligned}$$

Sketch the circle. Identify its center and radius. $$x^{2}-6 x+y^{2}+6 y+14=0$$

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 \(\frac{(y+4)^{2}}{\frac{1}{9}}-\frac{(x+3)^{2}}{\frac{1}{4}}=1\).

Find the center, vertices, foci, and eccentricity of the ellipse, and sketch its graph. Use a graphing utility to verify your graph. $$\frac{x^{2}}{16}+\frac{y^{2}}{81}=1$$

Use a graphing utility to graph the rotated conic. $$r=\frac{3}{1-\cos (\theta-\pi / 4)}$$

Identify and sketch the graph of the polar equation. Identify any symmetry and zeros of \(r .\) Use a graphing utility to verify your results. $$r=5 \cos 3 \theta$$

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(1.5,-2.82)$$

Use a graphing utility to graph the curve represented by the parametric equations. Use the graph and the Vertical Line Test to determine whether \(y\) is a function of \(x.\) $$\begin{aligned} &x=4 \sec \theta\\\ &y=2 \tan \theta \end{aligned}$$

Sketch the circle. Identify its center and radius. $$x^{2}-14 x+y^{2}+8 y+40=0$$

Find the standard form of the equation of the hyperbola, (b) find the center, vertices, foci, and asymptotes of the hyperbola, and (c) sketch the hyperbola. Use a graphing utility to verify your graph. \(4 x^{2}-9 y^{2}=36\)

Find the center, vertices, foci, and eccentricity of the ellipse, and sketch its graph. Use a graphing utility to verify your graph. $$\frac{(x-4)^{2}}{16}+\frac{(y+1)^{2}}{25}=1$$

Use a graphing utility to graph the rotated conic. $$r=\frac{7}{1+\sin (\theta-\pi / 3)}$$

Identify and sketch the graph of the polar equation. Identify any symmetry and zeros of \(r .\) Use a graphing utility to verify your results. $$r=\sin 5 \theta$$

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(-5.3,-0.78)$$

Use a graphing utility to graph the curve represented by the parametric equations. Use the graph and the Vertical Line Test to determine whether \(y\) is a function of \(x.\) $$\begin{aligned} &x=\sec \theta\\\ &y=\tan \theta \end{aligned}$$

Sketch the circle. Identify its center and radius. $$x^{2}+6 x+y^{2}-12 y+41=0$$

Find the standard form of the equation of the hyperbola, (b) find the center, vertices, foci, and asymptotes of the hyperbola, and (c) sketch the hyperbola. Use a graphing utility to verify your graph. \(25 x^{2}-4 y^{2}=100\)

Find the center, vertices, foci, and eccentricity of the ellipse, and sketch its graph. Use a graphing utility to verify your graph. $$\frac{(x+3)^{2}}{12}+\frac{(y-2)^{2}}{16}=1$$

Use a graphing utility to graph the rotated conic. $$r=\frac{4}{1-5 \cos (\theta+3 \pi / 4)}$$

Identify and sketch the graph of the polar equation. Identify any symmetry and zeros of \(r .\) Use a graphing utility to verify your results. $$r=-7 \sin 2 \theta$$

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(-7,0)$$

Use a graphing utility to graph the curve represented by the parametric equations. Use the graph and the Vertical Line Test to determine whether \(y\) is a function of \(x.\) $$\begin{aligned} &x=t / 2\\\ &y=\ln \left(t^{2}+1\right) \end{aligned}$$

Sketch the circle. Identify its center and radius. $$x^{2}+2 x+y^{2}-35=0$$

Find the standard form of the equation of the hyperbola, (b) find the center, vertices, foci, and asymptotes of the hyperbola, and (c) sketch the hyperbola. Use a graphing utility to verify your graph. \(6 y^{2}-3 x^{2}=24\)

Find the center, vertices, foci, and eccentricity of the ellipse, and sketch its graph. Use a graphing utility to verify your graph. $$\frac{(x+5)^{2}}{\frac{9}{4}}+(y-1)^{2}=1$$

Use a graphing utility to graph the rotated conic. $$r=\frac{9}{3-2 \cos (\theta+\pi / 2)}$$

Identify and sketch the graph of the polar equation. Identify any symmetry and zeros of \(r .\) Use a graphing utility to verify your results. $$r=3 \cos 4 \theta$$

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(0,-5)$$

Use a graphing utility to graph the curve represented by the parametric equations. Use the graph and the Vertical Line Test to determine whether \(y\) is a function of \(x.\) $$\begin{aligned} &x=10-0.01 e^{t}\\\ &y=0.4 t^{2} \end{aligned}$$

Sketch the circle. Identify its center and radius. $$x^{2}+y^{2}+10 y+9=0$$

Find the standard form of the equation of the hyperbola, (b) find the center, vertices, foci, and asymptotes of the hyperbola, and (c) sketch the hyperbola. Use a graphing utility to verify your graph. \(3 x^{2}-2 y^{2}=18\)

Find the center, vertices, foci, and eccentricity of the ellipse, and sketch its graph. Use a graphing utility to verify your graph. $$(x+2)^{2}+\frac{(y+4)^{2}}{\frac{1}{4}}=1$$

Use a graphing utility to graph the rotated conic. $$r=\frac{8}{4+3 \sin (\theta+\pi / 6)}$$

Identify and sketch the graph of the polar equation. Identify any symmetry and zeros of \(r .\) Use a graphing utility to verify your results. $$r=1+2 \sin \theta$$

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(1,1)$$

Determine how the plane curves differ from each other. (a) \(x=t\) \(y=2 t+1\) (b) \(x=\cos \theta\) \(y=2 \cos \theta+1\) (c) \(x=e^{-t}\) \(y=2 e^{-t}+1\) (d) \(x=e^{t}\) \(y=2 e^{t}+1\)

Find the \(x\) - and \(y\) -intercepts of the graph of the circle. $$(x+5)^{2}+(y-3)^{2}=25$$