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Problem 11.7.20
Hyperbolas and Eccentricity
In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.
y² − x² = 4
Problem 11.AAE.22
Polar Coordinates
Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.
e = 1/3, r sin θ = −6
Problem 11.PE.17
Lengths of Curves
Find the lengths of the curves in Exercises 13–19.
x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2
Problem 11.PE.12a
Finding Parametric Equations and Tangent Lines
Find parametric equations for the given curve.
Line through (1,-2) with slope 3
Problem 11.PE.12d
Finding Parametric Equations and Tangent Lines
Find parametric equations for the given curve.
9x² + 4y² = 36
Problem 11.PE.66
Graphing Conic Sections
Exercises 63-68 give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.
x²/169 + y²/144 = 1, right 5, up 12
Problem 11.PE.6
Identifying Parametric Equations in the Plane
Exercises 1–6 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 and indicate the direction of motion and the portion traced by the particle.
x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π
Problem 11.PE.76
Identifying Conic Sections
Complete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.
x² + y² + 4x + 2y = 1
Problem 11.PE.2
Identifying Parametric Equations in the Plane
Exercises 1–6 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 and indicate the direction of motion and the portion traced by the particle.
x = √t, y = 1 − √t, t ≥ 0
Problem 11.PE.24
Polar to Cartesian Equations
Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.
r cos (θ − 3π/4) = (√2)/2
Problem 11.PE.50
Area in Polar Coordinates
Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.
Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ
Problem 11.PE.58
Graphing Conic Sections
Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.
y² = −(8/3)x
Problem 11.PE.33
Cartesian to Polar Equations
Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.
x² + y² + 5y = 0
Problem 11.PE.62
Graphing Conic Sections
Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.
5y² − 4x² = 20
Problem 11.PE.54
Length in Polar Coordinates
Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.
r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2
Problem 11.3.3a
Polar Coordinates
Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.
a. (2, π/2)
Problem 11.1.29a
Finding Parametric Equations
Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².
a. once clockwise.
(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)
Problem 11.6.44a
Shifting Conic Sections
You may wish to review Section 1.2 before solving Exercises 39-56.
The hyperbola (y²/4) − (x²/5) = 1 is shifted 2 units down to generate the hyperbola (y + 2)²/4 − x²/5 = 1.
a. Find the center, foci, vertices, and asymptotes of the new hyperbola.
Problem 11.2.47a
Cycloid
a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).
Problem 11.6.74b
Theory and Examples
Volume Find the volume of the solid generated by revolving the region enclosed by the ellipse 9x² + 4y² = 36 about the y−axis.
Problem 11.3.7b
Cartesian to Polar Coordinates
Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.
b. (-3,0)
Problem 11.3.6c
Polar to Cartesian Coordinates
Find the Cartesian coordinates of the following points, given in polar coordinates.
c. (0, π/2)
Problem 11.3.9c
Cartesian to Polar Coordinates
Find the polar coordinates, 0 ≤ θ ≤ 2π and r ≤ 0, of the following points given in Cartesian coordinates.
c. (−1, √3)