Solutions for Rotation of Axes

Solutions to Try Its

1.

  • hyperbola
  • ellipse

2. [latex]\frac{{{x}^{\prime }}^{2}}{4}+\frac{{{y}^{\prime }}^{2}}{1}=1[/latex]

3.

  • hyperbola
  • ellipse

Solutions to Odd-Numbered Exercises

1. The [latex]xy[/latex] 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 [latex]xy[/latex] term.

7. [latex]AB=0[/latex], parabola

9. [latex]AB=-4<0[/latex], hyperbola

11. [latex]AB=6>0[/latex], ellipse

13. [latex]{B}^{2}-4AC=0[/latex], parabola

15. [latex]{B}^{2}-4AC=0[/latex], parabola

17. [latex]{B}^{2}-4AC=-96<0[/latex], ellipse

19. [latex]7{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}-4=0[/latex]

21. [latex]3{{x}^{\prime }}^{2}+2{x}^{\prime }{y}^{\prime }-5{{y}^{\prime }}^{2}+1=0[/latex]

23. [latex]\theta ={60}^{\circ },11{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}+\sqrt{3}{x}^{\prime }+{y}^{\prime }-4=0[/latex]

25. [latex]\theta ={150}^{\circ },21{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}+4{x}^{\prime }-4\sqrt{3}{y}^{\prime }-6=0[/latex]

27. [latex]\theta \approx {36.9}^{\circ },125{{x}^{\prime }}^{2}+6{x}^{\prime }-42{y}^{\prime }+10=0[/latex]

29. [latex]\theta ={45}^{\circ },3{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}-\sqrt{2}{x}^{\prime }+\sqrt{2}{y}^{\prime }+1=0[/latex]

31. [latex]\frac{\sqrt{2}}{2}\left({x}^{\prime }+{y}^{\prime }\right)=\frac{1}{2}{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}[/latex]

33. [latex]\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{8}+\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}=1[/latex]

35. [latex]\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}-\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{2}=1[/latex]

37. [latex]\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}[/latex]

39.

41.

43.

45.

47.

49.

51. [latex]\theta ={45}^{\circ }[/latex]

53. [latex]\theta ={60}^{\circ }[/latex]

55. [latex]\theta \approx {36.9}^{\circ }[/latex]

57. [latex]-4\sqrt{6}<k<4\sqrt{6}[/latex]

59. [latex]k=2[/latex]