Chapter 10

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=\cot \theta \csc \theta\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the inner loop of \(r=\cos \theta-\frac{1}{2}\)

Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of \(x\) and \(y .\) Answers are not unique. A circle centered at the origin with radius \(12,\) generated clockwise with initial point \((0,12)\)

Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{4}+\frac{y^{2}}{16}=1$$

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=2\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region outside the circle \(r=\frac{1}{2}\) and inside the circle \(r=\cos \theta\)

Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of \(x\) and \(y .\) Answers are not unique. A circle centered at \((2,3)\) with radius \(1,\) generated counterclockwise

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=3 \csc \theta\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the curve \(r=\sqrt{\cos \theta}\) and outside the circle \(r=1 / \sqrt{2}\)

Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of \(x\) and \(y .\) Answers are not unique. A circle centered at \((2,0)\) with radius \(3,\) generated clockwise

Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{5}+\frac{y^{2}}{7}=1$$

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=2 \sin \theta+2 \cos \theta\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the curve \(r=\sqrt{\cos \theta}\) and inside the circle \(r=1 / \sqrt{2}\) in the first quadrant

Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$12 x^{2}+5 y^{2}=60$$

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(\sin \theta=|\cos \theta|\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the right lobe of \(r=\sqrt{\cos 2 \theta}\) and inside the circle \(r=1 / \sqrt{2}\) in the first quadrant

Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of \(x\) and \(y .\) Answers are not unique. A circle centered at \((2,-4)\) with radius \(\frac{3}{2},\) generated counterclockwise with initial point \(\left(\frac{7}{2},-4\right)\)

Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse whose major axis is on the \(x\) -axis with length 8 and whose minor axis has length 6

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside one leaf of the rose \(r=\cos 5 \theta\)

Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse with vertices (±6,0) and foci (±4,0)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the rose \(r=4 \cos 2 \theta\) and outside the circle \(r=2\)

Find parametric equations that describe the circular path of the following objects. Assume \((x, y)\) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There is more than one way to describe any circle. The tip of the \(15 -inch\) second hand of a clock completes one revolution in 60 seconds.

Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse with vertices \((\pm 5,0),\) passing through the point \(\left(4, \frac{3}{5}\right)\)

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=8 \sin \theta\)

Make a sketch of the region and its bounding curves. Find the area of the region. The region inside the rose \(r=4 \sin 2 \theta\) and inside the circle \(r=2\)

Find parametric equations that describe the circular path of the following objects. Assume \((x, y)\) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There is more than one way to describe any circle. A bicyclist rides counterclockwise with constant speed around a circular velodrome track with a radius of \(50 \mathrm{m},\) completing one lap in 24 seconds.

Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci. An ellipse with vertices \((0,\pm 10),\) passing through the point \((\sqrt{3} / 2,5)\)

Convert the following equations to Cartesian coordinates. Describe the resulting curve. \(r=\frac{1}{2 \cos \theta+3 \sin \theta}\)

Find parametric equations that describe the circular path of the following objects. Assume \((x, y)\) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There is more than one way to describe any circle. A Ferris wheel has a radius of \(20 \mathrm{m}\) and completes a revolution in the clockwise direction at constant speed in 3 min. Assume that \(x\) and \(y\) measure the horizontal and vertical positions of a seat on the Ferris wheel relative to a coordinate system whose origin is at the low point of the wheel. Assume the seat begins moving at the origin.

Tabulate and plot enough points to sketch a graph of the following equations. \(r=8 \cos \theta\)

Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points. $$r=3 \sin \theta \text { and } r=3 \cos \theta$$

Find the slope of each line and a point on the line. Then graph the line. $$x=3+t, y=1-t$$

Tabulate and plot enough points to sketch a graph of the following equations. \(r=4+4 \cos \theta\)

Find the slope of each line and a point on the line. Then graph the line. $$x=4-3 t, y=-2+6 t$$

Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$\frac{x^{2}}{4}-y^{2}=1$$

Tabulate and plot enough points to sketch a graph of the following equations. \(r(\sin \theta-2 \cos \theta)=0\)

Find the slope of each line and a point on the line. Then graph the line. $$x=8+2 t, y=1$$

Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$\frac{y^{2}}{16}-\frac{x^{2}}{9}=1$$

Tabulate and plot enough points to sketch a graph of the following equations. \(r=1-\cos \theta\)

Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points. $$r=1 \text { and } r=\sqrt{2} \cos 2 \theta$$

Find the slope of each line and a point on the line. Then graph the line. $$x=1+2 t / 3, y=-4-5 t / 2$$

Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$4 x^{2}-y^{2}=16$$

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r=1-\sin \theta\)

In Exercises \(37-40,\) you found the intersection points of pairs of curves. Find the area of the entire region that lies within both of the following pairs of curves. $$r=3 \sin \theta \text { and } r=3 \cos \theta$$

Find a parametric description of the line segment from the point \(P\) to the point \(Q\). Solutions are not unique. $$P(0,0), Q(2,8)$$

Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$25 y^{2}-4 x^{2}=100$$

Graph the following equations. Use a graphing utility to check your work and produce a final graph. \(r=2-2 \sin \theta\)

In Exercises \(37-40,\) you found the intersection points of pairs of curves. Find the area of the entire region that lies within both of the following pairs of curves. $$r=2+2 \sin \theta \text { and } r=2-2 \sin \theta$$

Find a parametric description of the line segment from the point \(P\) to the point \(Q\). Solutions are not unique. $$P(1,3), Q(-2,6)$$

Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$\frac{x^{2}}{3}-\frac{y^{2}}{5}=1$$