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)

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

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)

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)

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)

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 left half of the parabola y=x ² +1, originating at (0, 1)

Spiral arc length Consider the spiral r=4θ, for θ≥0.

a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.