16. Parametric Equations & Polar Coordinates

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 segment of the parabola y=2x ²−4, where −1≤x≤5

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.

An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

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 line segment starting at P(0, 0) and ending at Q(2, 8)

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 horizontal line segment starting at P(8, 2) and ending at Q(−2, 2)

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=t,y= √(4−t²) a

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 = sin² t; 0 ≤ t ≤ π

Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 2 sinh t, y = 2 cosh t, −∞<t<∞

Write parametric equations for the rectangular equation below.

$x^2+y^2=25$

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=tan t, y=sec ² t−1 

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A set of parametric equations for a given curve is always unique.

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.

The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.

x = cos t, y = 8 sin t; t = π/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 line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x

7–8. Parametric curves and tangent lines

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

x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3

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)