22–23. Arc length Find the length of the following curves.

x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The circle x ² + y ² =9, generated clockwise

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

10–12. Parametric curves

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

x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.

x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

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

Polar conversion Consider the equation r=4/(sinθ+cosθ). 

a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.

x²/16 - y²/9 = 1; (20/3, -4)

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

e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.

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.

44–49. Areas of regions Find the area of the following regions.

The region inside the limaçon r=2+cosθ and outside the circle r=2

24–26. Sets in polar coordinates Sketch the following sets of points.

4 ≤ r² ≤ 9

51–52. {Use of Tech} Arc length of polar curves Find the approximate length of the following curves.

The limaçon r=3−6cosθ

53–57. Conic sections

d. Make an accurate graph of the curve.

x = 16y²

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

c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.

53–57. Conic sections

a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.

x = 16y²

A polar conic section Consider the equation r² = sec2θ

a. Convert the equation to Cartesian coordinates and identify the curve.

Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.

40–41. {Use of Tech} Slopes of tangent lines

b. Find the slope of the lines tangent to the curve at the origin (when relevant).

r = 1 −sin θ

53–57. Conic sections

c. Find the eccentricity of the curve.

y² - 4x² = 16

A polar conic section Consider the equation r² = sec2θ

b. Find the vertices, foci, directrices, and eccentricity of the curve."

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.

a. For what value of p is P tangent to H?

27–32. Polar curves Graph the following equations.

r = 3 cos 3θ

27–32. Polar curves Graph the following equations.

r = 3 sin 4θ

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

x² - y²/2 = 1

42–43. Intersection points Find the intersection points of the following curves.

r= √(cos3t) and r= √(sin3t)

Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.

53–57. Conic sections

a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.

x²/4 + y²/25 = 1

10–12. Parametric curves

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

x = t² + 4, y = -t, for -2 < t < 0; (5, 1)

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

4x² + 8y² = 16