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.

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)

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)

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.