16. Parametric Equations & Polar Coordinates

11–14. Working with parametric equations Consider the following parametric equations.

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

d. Describe the curve.

x=−t+6, y=3t−3; −5≤t≤5 

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

A circle centered at the origin with radius 4, generated counterclockwise

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

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)

Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2

c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

a. x = 1 + s, y = 2s and x = 1 + 2t, y = 3t

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.)

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

d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.

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

b. An object following the parametric curve x=2cos 2πt, y=2 sin 2πt circles the origin once every 1 time unit.

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

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