Polar Coordinates and Conic Sections: Get Stronger Answer Key

Understanding Polar Coordinates

  1. On the polar coordinate plane, a ray is drawn from the origin marking π/6 and a point is drawn when this line crosses the circle with radius 3.
  2. On the polar coordinate plane, a ray is drawn from the origin marking 7π/6 and a point is drawn when this line crosses the circle with radius 0, that is, it marks the origin.
  3. On the polar coordinate plane, a ray is drawn from the origin marking π/4 and a point is drawn when this line crosses the circle with radius 1.
  4. On the polar coordinate plane, a ray is drawn from the origin marking π/2 and a point is drawn when this line crosses the circle with radius 1.
  5. [latex]B\begin{array}{cc}\left(3,\dfrac{\text{-}\pi }{3}\right)\hfill & B\left(-3,\dfrac{2\pi }{3}\right)\hfill \end{array}[/latex]
  6. [latex]D\left(5,\dfrac{7\pi }{6}\right)D\left(-5,\dfrac{\pi }{6}\right)[/latex]
  7. [latex]\begin{array}{cc}\left(5,-0.927\right)\hfill & \left(-5,-0.927+\pi \right)\hfill \end{array}[/latex]
  8. [latex]\left(10,-0.927\right)\left(-10,-0.927+\pi \right)[/latex]
  9. [latex]\left(2\sqrt{3},-0.524\right)\left(-2\sqrt{3},-0.524+\pi \right)[/latex]
  10. [latex]\left(\begin{array}{cc}\text{-}\sqrt{3},\hfill & -1\hfill \end{array}\right)[/latex]
  11. [latex]\left(\begin{array}{cc}-\dfrac{\sqrt{3}}{2},\hfill & \dfrac{-1}{2}\hfill \end{array}\right)[/latex]
  12. [latex]\left(\begin{array}{cc}0,\hfill & 0\hfill \end{array}\right)[/latex]
  13. Symmetry with respect to the [latex]x[/latex]-axis, [latex]y[/latex]-axis, and origin.
  14. Symmetric with respect to [latex]x[/latex]-axis only.
  15. Symmetry with respect to [latex]x[/latex]-axis only.
  16. Line [latex]y=x[/latex]
  17. [latex]y=1[/latex]
  18. A hyperbola with vertices at (−4, 0) and (4, 0), the first pointing out into quadrants II and III and the second pointing out into quadrants I and IV.
    Hyperbola; polar form [latex]{r}^{2}\cos\left(2\theta \right)=16[/latex] or [latex]{r}^{2}=16\sec\theta[/latex].
  19. A straight line with slope 3 and y intercept −2.
    [latex]r=\dfrac{2}{3\cos\theta -\sin\theta }[/latex]
  20. A circle of radius 2 with center at (2, π/2).
    [latex]{x}^{2}+{y}^{2}=4y[/latex]
  21. A spiral starting at the origin and crossing θ = π/2 between 1 and 2, θ = π between 3 and 4, θ = 3π/2 between 4 and 5, θ = 0 between 6 and 7, θ = π/2 between 7 and 8, and θ = π between 9 and 10.
    [latex]x\tan\sqrt{{x}^{2}+{y}^{2}}=y[/latex]
  22. A cardioid with the upper heart part at the origin and the rest of the cardioid oriented up.
    [latex]y[/latex]-axis symmetry
  23. A cardioid with the upper heart part at the origin and the rest of the cardioid oriented down.
    [latex]y[/latex]-axis symmetry
  24. A rose with four petals that reach their furthest extent from the origin at θ = 0, π/2, π, and 3π/2.
    [latex]x[/latex]– and [latex]y[/latex]-axis symmetry and symmetry about the pole
  25. A rose with three petals that reach their furthest extent from the origin at θ = 0, 2π/3, and 4π/3.
    [latex]x[/latex]-axis symmetry
  26. The infinity symbol with the crossing point at the origin and with the furthest extent of the two petals being at θ = 0 and π.
    [latex]x[/latex]– and [latex]y[/latex]-axis symmetry and symmetry about the pole
  27. A spiral that starts at the origin crossing the line θ = π/2 between 3 and 4, θ = π between 6 and 7, θ = 3π/2 between 9 and 10, θ = 0 between 12 and 13, θ = π/2 between 15 and 16, and θ = π between 18 and 19.
    no symmetry

Area and Arc Length in Polar Coordinates

  1. [latex]\dfrac{9}{2}{\displaystyle\int }_{0}^{\pi }{\sin}^{2}\theta d\theta[/latex]
  2. [latex]32{\displaystyle\int }_{0}^{\frac{\pi}{2}}{\sin}^{2}\left(2\theta \right)d\theta[/latex]
  3. [latex]\dfrac{1}{2}{\displaystyle\int }_{\pi }^{2\pi }{\left(1-\sin\theta \right)}^{2}d\theta[/latex]
  4. [latex]{\displaystyle\int }_{{\sin}^{-1}\left(\frac{2}{3}\right)}^{\frac{\pi}{2}}{\left(2 - 3\sin\theta \right)}^{2}d\theta[/latex]
  5. [latex]{\displaystyle\int }{0}^{\pi }{\left(1 - 2\cos\theta \right)}^{2}d\theta -{\displaystyle\int }{0}^{\frac{\pi}{3}}{\left(1 - 2\cos\theta \right)}^{2}d\theta[/latex]
  6. [latex]4{\displaystyle\int }{0}^{\frac{\pi}{3}}d\theta +16{\displaystyle\int }{\frac{\pi}{3}}^{\frac{\pi}{2}}\left({\cos}^{2}\theta \right)d\theta[/latex]
  7. [latex]9\pi[/latex]
  8. [latex]\dfrac{9\pi }{4}[/latex]
  9. [latex]\dfrac{9\pi }{8}[/latex]
  10. [latex]\dfrac{18\pi -27\sqrt{3}}{2}[/latex]
  11. [latex]\dfrac{4}{3}\left(4\pi -3\sqrt{3}\right)[/latex]
  12. [latex]\dfrac{3}{2}\left(4\pi -3\sqrt{3}\right)[/latex]
  13. [latex]2\pi -4[/latex]
  14. [latex]{\displaystyle\int }_{0}^{2\pi }\sqrt{{\left(1+\sin\theta \right)}^{2}+{\cos}^{2}\theta }d\theta[/latex]
  15. [latex]\sqrt{2}{\displaystyle\int }_{0}^{1}{e}^{\theta }d\theta[/latex]
  16. [latex]\dfrac{\sqrt{10}}{3}\left({e}^{6}-1\right)[/latex]
  17. [latex]32[/latex]
  18. [latex]6.238[/latex]
  19. [latex]2[/latex]
  20. [latex]4.39[/latex]
  21. [latex]A=\pi {\left(\dfrac{\sqrt{2}}{2}\right)}^{2}=\dfrac{\pi }{2}\text{ and }\dfrac{1}{2}{\displaystyle\int }_{0}^{\pi }\left(1+2\sin\theta \cos\theta \right)d\theta =\dfrac{\pi }{2}[/latex]
  22. [latex]C=2\pi \left(\dfrac{3}{2}\right)=3\pi \text{ and }{\displaystyle\int }_{0}^{\pi }3d\theta =3\pi[/latex]
  23. [latex]C=2\pi \left(5\right)=10\pi \text{ and }{\displaystyle\int }_{0}^{\pi }10d\theta =10\pi[/latex]
  24. [latex]\dfrac{dy}{dx}=\dfrac{{f}^{\prime }\left(\theta \right)\sin\theta +f\left(\theta \right)\cos\theta }{{f}^{\prime }\left(\theta \right)\cos\theta -f\left(\theta \right)\sin\theta }[/latex]
  25. The slope is [latex]\dfrac{1}{\sqrt{3}}[/latex].
  26. The slope is [latex]0[/latex].
  27. At [latex]\left(4,0\right)[/latex], the slope is undefined. At [latex]\left(-4,\dfrac{\pi }{2}\right)[/latex], the slope is 0.
  28. The slope is undefined at [latex]\theta =\dfrac{\pi }{4}[/latex].
  29. Slope =[latex]−1[/latex].
  30. Slope is [latex]\dfrac{-2}{\pi }[/latex].
  31. Calculator answer: [latex]−0.836.[/latex]
  32. Horizontal tangent at [latex]\left(\pm\sqrt{2},\dfrac{\pi }{6}\right)[/latex], [latex]\left(\pm\sqrt{2},-\dfrac{\pi }{6}\right)[/latex].
  33. Horizontal tangents at [latex]\dfrac{\pi }{2},\dfrac{7\pi }{6},\dfrac{11\pi }{6}[/latex]. Vertical tangents at [latex]\dfrac{\pi }{6},\dfrac{5\pi }{6}[/latex] and also at the pole [latex]\left(0,0\right)[/latex].

Conic Sections

  1. [latex]{y}^{2}=16x[/latex]
  2. [latex]{x}^{2}=2y[/latex]
  3. [latex]{x}^{2}=-4\left(y - 3\right)[/latex]
  4. [latex]{\left(x+3\right)}^{2}=8\left(y - 3\right)[/latex]
  5. [latex]\dfrac{{x}^{2}}{16}+\dfrac{{y}^{2}}{12}=1[/latex]
  6. [latex]\dfrac{{x}^{2}}{13}+\dfrac{{y}^{2}}{4}=1[/latex]
  7. [latex]\dfrac{{\left(y - 1\right)}^{2}}{16}+\dfrac{{\left(x+3\right)}^{2}}{12}=1[/latex]
  8. [latex]\dfrac{{x}^{2}}{16}+\dfrac{{y}^{2}}{12}=1[/latex]
  9. [latex]\dfrac{{x}^{2}}{25}-\dfrac{{y}^{2}}{11}=1[/latex]
  10. [latex]\dfrac{{x}^{2}}{7}-\dfrac{{y}^{2}}{9}=1[/latex]
  11. [latex]\dfrac{{\left(y+2\right)}^{2}}{4}-\dfrac{{\left(x+2\right)}^{2}}{32}=1[/latex]
  12. [latex]\dfrac{{x}^{2}}{4}-\dfrac{{y}^{2}}{32}=1[/latex]
  13. [latex]e=1[/latex], parabola
  14. [latex]e=\dfrac{1}{2}[/latex], ellipse
  15. [latex]e=3[/latex], hyperbola
  16. [latex]r=\dfrac{4}{5+\cos\theta }[/latex]
  17. [latex]r=\dfrac{4}{1+2\sin\theta }[/latex]
  18. Hyperbola
  19. Ellipse
  20. Ellipse