Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.

r cos θ = -4

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.

r = 1/(2 - 2 sin θ)

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

x²/9 + y²/4 = 1

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.

r = 3/(2 + cos θ)

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola with focus at (3, 0)

Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

A hyperbola with vertices (±4, 0) and foci (±6, 0)

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the curve r = √(cos θ)

15–22. Sets in polar coordinates Sketch the following sets of points.

r = 3

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 4 cos 2θ; at the tips of the leaves

Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)

A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 4 + sin θ; (4, 0) and (3, 3π/2)

53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, 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 are many ways to describe any circle.

A bicyclist rides counterclockwise with constant speed around a circular velodrome track with a radius of 50 m, completing one lap in 24 seconds.

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.

(-4, 3π/2)

81–88. Arc length Find the arc length of the following curves on the given interval.

x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola that opens to the right with directrix x = -4

Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.

x = 4 cos t, y = 4 sin t; slope = 1/2

What is the polar equation of the horizontal line y = 5?

90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

Let L be the latus rectum of the parabola y ² =4px for p>0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|FP|+ is a constant. Find the constant.

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.

x=cos t+t sin t,y=sin t−t cos t; t=π/4

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = 8 + 2t, y = 1; −∞ < t < ∞

{Use of Tech} Implicit function graph Explain and carry out a method for graphing the curve x = 1 + cos² y − sin² y using parametric equations and a graphing utility.

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = r − 1, y = r³; −4 ≤ r ≤ 4