Explain why the slope of the line θ=π/2 is undefined.

Circles in general Show that the polar equation

r² - 2r r₀ cos(θ - θ₀) = R² - r₀²

describes a circle of radius R whose center has polar coordinates (r₀, θ₀)

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/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 the parabola y ² =4px or x ² =4py is 4|p|.

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

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

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 = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π

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)

Air drop A plane traveling horizontally at 80 m/s over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by

x = 80t, y = −4.9t² + 3000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release (see figure). Graph the trajectory of the packet and find the coordinates of the point where the packet lands.

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

x = 2 cos t, y = 8 sin t; slope = -1

Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2. 

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

63–74. Arc length of polar curves Find the length of the following polar curves.

The spiral r = θ², for 0 ≤ θ ≤ 2π

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

x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π

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)

How does the eccentricity determine the type of conic section?

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.

y = 3

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)

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

x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π

29–32. Intersection points 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 = 2 cos θ and r = 1 + cos θ

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

The region inside one leaf of r = cos 3θ

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

(-4, 4√3)

45–60. Areas of regions Find the area of the following regions.

The region common to the circles r = 2 sin θ and r = 1

Use calculus to find the arc length of the line segment x=3t+1, y=4t, for 0≤t≤1. Check your work by finding the distance between the endpoints of the line segment. 

102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ>0. Let a=1 and a=−1.

Spiral of Archimedes: r = aθ

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.

(1, 2π/3)

Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge? 

{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.

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.

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

An ellipse with vertices (±5, 0), passing through the point (4, 3/5)