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문제 12.R.23
22–23. Arc length Find the length of the following curves.
x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4
문제 12.R.17
14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The circle x ² + y ² =9, generated clockwise
문제 12.R.56c
53–57. Conic sections
c. Find the eccentricity of the curve.
x²/4 + y²/25 = 1
문제 12.R.12a
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)
문제 12.R.4
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
문제 12.R.7a
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
문제 12.R.36a
Polar conversion Consider the equation r=4/(sinθ+cosθ).
a. Convert the equation to Cartesian coordinates and identify the curve it describes.
문제 12.R.58
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)
문제 12.R.1e
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.
문제 12.R.1a
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.
문제 12.R.46
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
문제 12.R.26
24–26. Sets in polar coordinates Sketch the following sets of points.
4 ≤ r² ≤ 9
문제 12.R.51
51–52. {Use of Tech} Arc length of polar curves Find the approximate length of the following curves.
The limaçon r=3−6cosθ
문제 12.R.54d
53–57. Conic sections
d. Make an accurate graph of the curve.
x = 16y²
문제 12.R.1c
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.
문제 12.R.54a
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x = 16y²
문제 12.R.60a
A polar conic section Consider the equation r² = sec2θ
a. Convert the equation to Cartesian coordinates and identify the curve.
문제 12.R.35
Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.
문제 12.R.41b
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
문제 12.R.55c
53–57. Conic sections
c. Find the eccentricity of the curve.
y² - 4x² = 16
문제 12.R.60b
A polar conic section Consider the equation r² = sec2θ
b. Find the vertices, foci, directrices, and eccentricity of the curve."
문제 12.R.72a
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?
문제 12.R.29
27–32. Polar curves Graph the following equations.
r = 3 cos 3θ
문제 12.R.31
27–32. Polar curves Graph the following equations.
r = 3 sin 4θ
문제 12.R.53b
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
x² - y²/2 = 1
문제 12.R.43
42–43. Intersection points Find the intersection points of the following curves.
r= √(cos3t) and r= √(sin3t)
문제 12.R.70
Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.
문제 12.R.56a
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x²/4 + y²/25 = 1
문제 12.R.10a
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)
문제 12.R.57b
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
4x² + 8y² = 16