Chapter 10

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. \(9 x^{2}-y^{2}-36 x-6 y+18=0\)

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$x^{2}+9 y^{2}=36$$

Use a graphing utility to graph the rotated conic. $$r=\frac{10}{3+9 \sin (\theta+2 \pi / 3)}$$

Identify and sketch the graph of the polar equation. Identify any symmetry and zeros of \(r .\) Use a graphing utility to verify your results. $$r=\sqrt{3}-2 \cos \theta$$

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(-3,-3)$$

Determine how the plane curves differ from each other. (a) \(x=2 \sqrt{t}\) \(y=4-\sqrt{t}\) (b) \(x=2 \sqrt[3]{t}\) \(y=4-\sqrt[3]{t}\) (c) \(x=2(t+1)\) \(y=3-t\) (d) \(x=-2 t^{2}\) \(y=4+t^{2}\)

Find the \(x\) - and \(y\) -intercepts of the graph of the circle. $$(x-1)^{2}+(y+4)^{2}=16$$

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. \(x^{2}-9 y^{2}+36 y-72=0\)

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$16 x^{2}+y^{2}=16$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Parabola} &e=1&x=-1\end{array}$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=8 \cos 2 \theta$$

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(-\sqrt{3},-\sqrt{3})$$

Eliminate the parameter and obtain the standard form of the rectangular equation. Line through \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\): \(x=x_{1}+t\left(x_{2}-x_{1}\right)\) \(y=y_{1}+t\left(y_{2}-y_{1}\right)\)

Find the \(x\) - and \(y\) -intercepts of the graph of the circle. $$(x-6)^{2}+(y+3)^{2}=16$$

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. \(2 x^{2}-7 y^{2}+16 x+18=0\)

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$49 x^{2}+4 y^{2}-196=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Parabola} &e=1&y=-4\end{array}$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=-\cos 2 \theta$$

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(\sqrt{3},-1)$$

Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: \(x=h+r \cos \theta, y=k+r \sin \theta\)

Find the \(x\) - and \(y\) -intercepts of the graph of the circle. $$(x+7)^{2}+(y-8)^{2}=4$$

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. \(3 y^{2}-5 x^{2}+6 y-60 x-192=0\)

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$4 x^{2}+49 y^{2}-196=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Ellipse} &e=\frac{1}{2}&y=1\end{array}$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=2(5-\sin \theta)$$

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(-3,4)$$

Eliminate the parameter and obtain the standard form of the rectangular equation. Ellipse: \(x=h+a \cos \theta, y=k+b \sin \theta\)

Find the \(x\) - and \(y\) -intercepts of the graph of the circle. $$x^{2}-2 x+y^{2}-6 y-27=0$$

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. \(9 y^{2}-x^{2}+2 x+54 y+62=0\)

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$9 x^{2}+4 y^{2}+36 x-24 y+36=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Ellipse} &e=\frac{3}{4}&y=-4\end{array}$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=6-4 \sin \theta$$

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(3,-1)$$

Eliminate the parameter and obtain the standard form of the rectangular equation. Hyperbola: \(x=h+a \sec \theta, y=k+b \tan \theta\)

Find the \(x\) - and \(y\) -intercepts of the graph of the circle. $$x^{2}+8 x+y^{2}+2 y+9=0$$

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. 9 x^{2}-y^{2}+54 x+10 y+55=0

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$9 x^{2}+4 y^{2}-54 x+40 y+37=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Hyperbola} &e=2&x=1\end{array}$$

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$(3,-2)$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=\frac{3}{\sin \theta-2 \cos \theta}$$

earthquake was felt up to 52 miles from its epicenter. You were located 40 miles west and 30 miles south of the epicenter. (a) Let the epicenter be at the point \((0,0) .\) Find the standard equation that describes the outer boundary of the earthquake. (b) Would you have felt the earthquake? (c) Verify your answer to part (b) by graphing the equation of the outer boundary of the earthquake and plotting your location. How far were you from the outer boundary of the earthquake?

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (±1,0)\(;\) asymptotes: \(y=\pm 5 x\)

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$6 x^{2}+2 y^{2}+18 x-10 y+2=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Hyperbola} &e=\frac{3}{2}&x=-1\end{array}$$

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$(-5,2)$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=\frac{6}{2 \sin \theta-3 \cos \theta}$$

Use the results of Exercises 37-40 to find a set of parametric equations for the line or conic. Circle: center: \((-2,-5)\); radius: 7

A landscaper has installed a circular sprinkler that covers an area of 2000 square feet. (a) Find the radius of the region covered by the sprinkler. Round your answer to three decimal places. (b) The landscaper increases the area covered to 2500 square feet by increasing the water pressure. How much longer is the radius?

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (0,±3)\(;\) asymptotes: \(y=\pm 3 x\)

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$x^{2}+4 y^{2}-6 x+20 y-2=0$$