65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

53–57. Conic sections

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

x² - y²/2 = 1

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

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

4x² + 8y² = 16

53–57. Conic sections

d. Make an accurate graph of the curve.

x = 16y²

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

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

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?

Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?

b. r=(½)+sinθ

General equations for a circle Prove that the equations  

X = a cos t + b sin t,  y = c cos t + d sin t  

where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0. 

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.

r = 3/(1 - 2 cos θ)

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

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

The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ

A polar conic section Consider the equation r² = sec2θ

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

19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve

The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.

x=cos t+t sin t,y=sin t−t cos t; t=π/4

Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2. 

Use calculus to find the arc length of the line segment x=3t+1, y=4t, for 0≤t≤1. Check your work by finding the distance between the endpoints of the line segment. 

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

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).

x=2 t,y=3t−4;−10≤d≤10 

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

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).

x=−t+6, y=3t−3; −5≤t≤5 

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 = 3 + t, y = 1 − t; 0 ≤ t ≤ 1

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 = √t + 4, y = 3√t; 0 ≤ t ≤ 16

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 = 3 cos t, y = 3 sin t; π ≤ t ≤ 2π

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

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 = 1 + sin t; 0 ≤ t ≤ 2π

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

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 = 8 + 2t, y = 1; −∞ < t < ∞

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

x=2 sin 8t, y=2 cos 8t 

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

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 

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=2 t,y=3t−4;−10≤t≤10