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Study Guides > Precalculus II

Solutions for Rotation of Axes

Solutions to Try Its

1.
  • hyperbola
  • ellipse
2. x′24+y′21=1\frac{{{x}^{\prime }}^{2}}{4}+\frac{{{y}^{\prime }}^{2}}{1}=1 3.
  • hyperbola
  • ellipse

Solutions to Odd-Numbered Exercises

1. The xyxy term causes a rotation of the graph to occur. 3. The conic section is a hyperbola. 5. It gives the angle of rotation of the axes in order to eliminate the xyxy term. 7. AB=0AB=0, parabola 9. AB=−4<0AB=-4<0, hyperbola 11. AB=6>0AB=6>0, ellipse 13. B2−4AC=0{B}^{2}-4AC=0, parabola 15. B2−4AC=0{B}^{2}-4AC=0, parabola 17. B2−4AC=−96<0{B}^{2}-4AC=-96<0, ellipse 19. 7x′2+9y′2−4=07{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}-4=0 21. 3x′2+2x′y′−5y′2+1=03{{x}^{\prime }}^{2}+2{x}^{\prime }{y}^{\prime }-5{{y}^{\prime }}^{2}+1=0 23. θ=60∘,11x′2−y′2+3x′+y′−4=0\theta ={60}^{\circ },11{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}+\sqrt{3}{x}^{\prime }+{y}^{\prime }-4=0 25. θ=150∘,21x′2+9y′2+4x′−43y′−6=0\theta ={150}^{\circ },21{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}+4{x}^{\prime }-4\sqrt{3}{y}^{\prime }-6=0 27. θ≈36.9∘,125x′2+6x′−42y′+10=0\theta \approx {36.9}^{\circ },125{{x}^{\prime }}^{2}+6{x}^{\prime }-42{y}^{\prime }+10=0 29. θ=45∘,3x′2−y′2−2x′+2y′+1=0\theta ={45}^{\circ },3{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}-\sqrt{2}{x}^{\prime }+\sqrt{2}{y}^{\prime }+1=0 31. 22(x′+y′)=12(x′−y′)2\frac{\sqrt{2}}{2}\left({x}^{\prime }+{y}^{\prime }\right)=\frac{1}{2}{\left({x}^{\prime }-{y}^{\prime }\right)}^{2} 33. (x′−y′)28+(x′+y′)22=1\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{8}+\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}=1 35. (x′+y′)22−(x′−y′)22=1\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}-\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{2}=1 37. 32x′−12y′=(12x′+32y′−1)2\frac{\sqrt{3}}{2}{x}^{\prime }-\frac{1}{2}{y}^{\prime }={\left(\frac{1}{2}{x}^{\prime }+\frac{\sqrt{3}}{2}{y}^{\prime }-1\right)}^{2} 39. 41. 43. 45. 47. 49. 51. θ=45∘\theta ={45}^{\circ } 53. θ=60∘\theta ={60}^{\circ } 55. θ≈36.9∘\theta \approx {36.9}^{\circ } 57. −46<k<46-4\sqrt{6}<k<4\sqrt{6} 59. k=2k=2

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